They tried it and apparently weren't impressed with the results, so it wasn't something that they pursued on a large scale.
In the larger picture, we can see that a great deal of time, money, labor and material went into exploring alternatives for the Fw190. This holds true with the Bf109 and other production types. So on the one hand, the He112 could have been upgraded or enhanced but on the otherhand, how many prototypes would it take to hit on the right combination?
Look at the time consumed in trying to salvage the Me210 when it should have been shelved and either go back to the drawing board or move on to the next, more promising type. There were over 24 prototypes of the Me210 that spanned 4 years: V1 1939 - V24 1943. While this may be seen as the natural course of development for a type, it is a huge setback during wartime, especially when things aren't going well for the Fatherland.
And just in case anyone's wondering, there were over 80 prototypes for the Fw190...
All good points, though as far as Heinkel aircraft development was concerned, that implies investing a bit more on competing single engine fighters to complement the 109 and 190 might have avoided the need for such exhaustive development investment across those other types.
With the 109 you have the 112 in the same timeframe, but the He 100 could/should have totally displaced further He 112 development and offered a competitor/complement to the Fw 190 program with closer manufacturing cost to the 109.
The He 100 should have retained the He 112's strengths as best as possible while incorporating newer engineering and design and more efficient manufacturing. The problem seems to have been both political preference against Heinkel (and towards Messerschmitt) and Heinkel focusing on experimental high-speed design and speed-record setting subtypes than a practical, efficient military aircraft design. Supposedly abandoning further development when the Jumo 211 became the only remotely viable powerplant available further accentuates this. (a decently flexibly engineered He 100 should have already been testing Jumo 211 powered variants as well as fully conventional radiator arrangements) In fact, supporting the Jumo 211 would have necessitated conventional radiator development as no variants were available with pressurized evaporative cooling compatible cooling systems. (the 211F introduced pressurized cooling somewhat later and would have allowed more compact/low drag radiator designs as well, but not for early testing, and probably well after surface cooling systems had been proven unworkable)
They didn't need an aircraft absolutely cheaper to manufacture than the Bf 109, but they needed one attractive enough all around to supplement or replace it. (total cost from production to service in the field -both handling/performance qualities and reliability/maintenance qualities)
Heinkel's wooden wing pet project might have been a smarter idea to continue development of (compared to his speed record pet projects) even if the RLM didn't like the idea, having the He 100 at least designed with wooden wing variants tested and cited for manufacturing cost and overall serviceability would have been significant. (a modular wing design able to accept either metal or wooden wings in the field would have been excellent if it could be done at reasonable cost and reliability)
If continued He 112 development could have met all those qualities better than the He 100 program ended up doing, then yes that's an extremely good argument for the He 112.
The "potential" thing is ripe for practical argument/discussion. You can probably "fix" the faults of MOST airplanes. The real question is, "is it economically feasible to do so?" A fully "tweaked" He 112 might have been a really great prototype with performance to spare ... and the RLM still might not have bought it, so you have to decide if the potential investment is worth the risk. I've always wondered why none of Germany's allies bought it in quantity, and there is likely a good reason, potential development notwithstanding. They apparently built 104 of them, which would not be profitable.
Retaining the thicker wing of the 112 might not have been a bad thing. It LOOKS thick, but without any definitive confirmation on the airfoil section used, it's hard to tell if it's using a thick airfoil or just a thick, low aspect ratio wing with relatively long chord. Additionally, one solution for improving high speed drag would be to extend the leading and/or trailing edges of the wing to conform to a thinner airfoil shape at the expense of increasing area and decreasing aspect ratio -if the wing root is the only section using a particularly high thickness to chord ratio, having highly tapered wing root extensions would be an even simpler modification, akin to what the Me 262 HG-I adopted.
Low aspect ratio is bad for low speed lift to drag ratio, but the elliptical shape compensates for that while low aspect ratio is good for strong wings with high roll rate and potential to exploit a thin airfoil stretched to a relatively decent area and internal storage capacity . (honestly it's probably something more post-war straight-winged jets should have attempted short of adopting swept wings, but that's another topic)
The elliptical shape might have increased manufacturing cost/time unless smart engineering workaround for that were found as well. (I'm not sure if that is as true for the wooden wing or not)
A heavily tapered straight wing is a good compromise (the He 100 and especially He 162 did such), as is an elliptical wing with a straight leading edge (like Republic tended to use, and Heinkel used on the He 178 and He 280, very thin and wooden in the former's case). The He 162's wing is a good example of physically thick, low aspect ratio with a thin airfoil section used and high internal capacity. (that's the sort of engineering they should have been focusing on pre-war, granted I believe the He 162 -like most jets- dispensed with self-sealing concerns entirely)
If the wing alone was one of the more costly elements of the 112, particularly the section outboard of the gear attachments (and gull bend), then replacing that entirely with a lower cost compromise (like a highly tapered straight wing) might have been best. Having the wider spaced gear that still retracted outward like the 109 should have allowed detachable outer wing sections (potentially modular wood or metal installation too), so that could have been one area to focus on design streamlining as well. (and one of the areas the He 100's inward retracting gear might not be an improvement, especially if the thick wing center section provided space for both fuel and machine guns -and potentially cannons once the MG 151 became available)
The Bf 109 was ripe with faults but, for all it's faults, was a VERY good offensive attack fighter that was fairly cheap to make given the level of performance it achieved ... which was near or at the top of the field. It is quite possible the DB-powered He 112 had every bit of the performance of the Bf 109 at a greater cost, and never got a big order simply due to the price difference. Or ... it may have NOT had the performance the Bf 109 had, and was STILL more expensive.
One advantage of the He 112 not being in service and mass production (compared to the 109) would mean more substantial design changes and fixes without disrupting existing production lines. Where the 109's refinements were curtailed in favor of maintained volume production, the He 112 (or 100) could have delved unimpeded until actual mass production commenced.
It's also not just 109 costs that must be considered, but Fw 190 costs too ... the 190 was a significantly more expensive aircraft to manufacture and a cheaper (but not 109 cheap) competitor might have complemented it well to the point of proving its other qualities were strong enough to displace the 109's raw manufacturing costs. (from raw performance to versitility in operation, to operational costs -material and man hours- to material costs going into manufacturing -like a wooden wing that might not make the aircraft's price tag any lower but reduced strategic material usage -ironically by the time wooden combat aircraft design requirements came up mid/late war, they ran into shortages of the right sort of wood, craftsmen, equipped factory space, and particularly glue to manufacture reliable aircraft components whereas developing that early war would have paid off much more -contracting some of Gotha's extensive wooden construction likely would have been wise as well, industry wide at least if not for Heinkel specifically given his company's and engineers' more existing expertise there)
For instance, I think the AR 240 had potential, and it was fast enough that if Arado couldn't fix the stability, I bet a rival firm could have and then the Germans would have HAD their "Mosquito" or at least close to it ... but it never happened for real.
You have to appreciate/marvel the potential that went untapped in the German aircraft industry ranging from the Me 264 to the Fw 191 to the AR 240 and so on down a long list of "might-have-beens" up to and including the BV 155 high-altitude version of the Bf 109. The potential was amazing in a lot of cases.
Factual evidence is mostly circumstantial from what I've seen, but the case of the Ar 240 really seems to point to RLM prejudice towards the Me 210 project and lack of interest or resources being fed to the 240, particularly given its relatively early development dead-end. (there was the larger Ar 440 development, but maintaining focus on the original 240 and refinement thereof seems like it would have matured much more quickly, especially had more powerful engines -realistic engines like the DB-605, 213, and 603, or BMW 801, not the likes of the Jumo 222)
From personal experience, the Bf 109 is NOT simple to work on. There are many things that WOULD be simple if the order of assembly was followed but, once the aircraft is together, you don't have the luxury of order of assembly. For instance, if the landing gear attach points are being assembled onto a NEW Bf 109, they are easy. If the Bf 109 is assembled, then someone is down in the cockpit, head first, in a very uncomfortable and awkward position trying to remove and reinstall the bolts that hold the brackets on while the rudder pedals and other items are all in the way. We finally said enough and removed all the pieces ... and, in fact, completely disassembled the cockpit for access, including removing the instraument panel.
Hehe that was exactly one of the contention points between the 109 and 112, the Messer was designed for ease of production, the Heinkel for ease of maintenance, in both cases the non-prioritized attribute suffered by comparison...
Given notes on the He 100's access panels and cramped engine installation in general (let alone the horribly complex pump system in the surface cooled systems), Heinkel definitely didn't maintain the same focus on ease of serviceability and maintenance with the He 100 as the He 112 ... another area where the He 100 could or should have been better but failed. (the main reason to go for a new design rather than further 112 development was to comprehensively fix more fundamental design flaws of the 112 and allow new technology to be introduced from the ground up ... this doesn't appear to have eventuated)
Silly of me, I completely forgot that Heinkel went on to design the He-100 and that in its final version it did feature a radiator for a DB 601 installation, so perhaps hey would have gone with some larger version of its retractable radiator?
Anyway, I think the only shot this aircraft would have had is if the KM carriers had been completed, one of the A-1s was used a a prototype for a carrier version and with its wider landing gear it would have been a better proposition than the 109T.
There was a lot of discussion on this and information thrown back and forth (and attempted deduction based on limited information and conflicting incomplete accounts of testing), and the results were rather ambiguous as far as the final (supposed D-1) variant of the He 100 is concerned.
That said, there was more definitive conclusions on Heinkel's retractable radiator being a dead-end compared to low-drag wing mounted radiators with boundary layer bleed plates/ducting (that might not be the right terminology, but the context was the sort of low-drag radiators used in the 109F and Spitfire V -or might have been IX- as well as the Mosquito among others). The two major alternatives were nose mounted radiators (or annular Junkers style ones) which traded low drag for low weight and compact, reliable, easier to maintain, less vulnerable installation. (the third option was an even more optimized low drag ducted radiator installation similar to what the Mustang implemented or short of that other fuselage/wing-center buried radiator modules like the P-39 used)
Given the He 100 also relied on a wet wing (which would have had fuel capacity cut when switching to self-sealing fuel cells) and the relatively small wing likely needing to be enlarged for further development with larger engines (DB601E or Jumo 211F), switching to a new wing with embedded radiators might have been the better route for further development. (omitting the fuselage radiator in favor of fuel tankage) Either that or a refined, low-drag fixed-position fuselage radiator and redesigned wing suited to self sealing tanks. (the wing-root embedded gun locations would be one of the bigger reasons for avoiding wing-mounted radiators, though the two might not have been mutually exclusive if designed with both in mind)
And yes, the He 112 likely would have been better suited to carrier use as well (more so with extended wings), and better take-off and landing performance than the 109 in general. (though to be totally fair, adding wing root extensions like the 109H proposed would have probably been a better solution than the 109T as well, particularly with a folding mechanism included for the outer wings and fuel tankage increased in that nice thick wing root extension region)