I've been meaning to comment on a bunch of points in this discussion for months now, but kept getting distracted, sooo, here goes.
[+] ability to relight in flight (Seems trivial but THIS WAS A BIG CONCERN. -the primary reason to give early jet A/C so large wing areas and good low speed handling -that if engine was lost during take off/landing it doesn´t critically effect the survivability of the pilot)
the first jet engine which adressed most of these points -to my knowledge- was the BMW-003A1. The BMW- jet engine project took longer than Junkers -004 but it was a more matured design. Lighter and smaller, better thrust-weight ratio, 150 hours certified lifetime for the hot turbine section (the compressor section had a significantly larger lifetime), an accelerator valve to prevent the burn out of the turbine blade due to rapid throttle changes (AFAIK, this was the first jet engine, whiches throttles could be less gingerly advanced and returned without fear of damaging the engine), good altitude performance with very few documented compressor stalls (could be relighted in flight) and overrew capability for increased thrust. There was one aspect which was not included, an automatic exhoust jet needle controll such as employed by Junkers, requiring the operator to controll this aspect.
I've gotten a lot of mixed information on these issues and actually had the impression the 004 series (or at least the most common service models) were capable of being re-started from inside the cockpit, though maybe more variables complicated this. (oddly, some flight sims seem to model the reverse of what you show: 004s able to be re-started vs 003s being totally dead on flame-out - the Il-2 series seems to do this fairly consistently)
On a side note, with the 003 having so many advantages, it's a bit odd there were no serious considerations for producting an Me-262 variant with those engines. (probably a fair argument that slating those engines for thatpurpose would have made more sense than reserving them for the likes of the He-162 ... including possibly making the Me 262 easier to fly -more fool-proof engines and throttle control, lower overall weight, lower stall speed, higher roll rate, longer endurance/range and higher top speed and climb rate -at very least if including emergency overrev thrust)
Radial compressor jet engines like those mentioned previously were probably better suited for the low thrust ratings concerned in ww2 but required more machining, milling and higher grade steel ressources.
This is true for some of the more elaborate machined impellers, but Heinkel appears to have been using a sheet metal composite construction for both their compressors and turbines (and a large portion of the diffusor and combustion chamber sections too). The compressor of the HeS 3, 6, and 8 being made from a steel hub with aluminum blades/vanes mated to it. (the radial turbine was similar but all steel, I believe Krupp stainless steel similar to the Tinadur alloy Junkers adopted)
There's a lot of nice info on this here:
ASME DC | Proceedings | GT1999 | Volume 1: Aircraft Engine; Marine; Turbomachinery; Microturbines and Small Turbomachinery | Pioneering Turbojet Developments of Dr. Hans von Ohain â€" From the HeS 1 to the HeS 011
(which I'm glad they've finally made free to public access -years ago a big chunk was viewable though google's cache, but hasn't for quite some time ... sans paying the $25 fee they were asking)
The requirement to produce only jet engines with spare free charakteristics would preclude the idea that the Luftwaffe could have fielded a working 4000 to 5000lbs jet engine in time for ww2, be it radial or axial design. As I mentioned previously, this requirement set back the whole jet engine project by approx. 1.5 years. Many of the issues encountered historically really just need to be adressed in order to be able to move beyond this low thrust rating to higher performance engines.
I'll mostly agree here, with the exception that ~5000 lb engines of relatively primitive construction might have been possible in the time frame, but too bulky to be all that useful. The 3000~4000 range is more seeable. A german equivalent of something as conservative as the Halford H.1/Goblin would have been interesting ... or scaled up a bit larger. (the later Ghost was still pretty simple ... also much heavier than comparable Whittle-based designs at Rolls Royce and GE -that and a Ghost size engine probably would have been more in the 3200~3600 lb thrust range given the thrust the Goblin itself was putting out at the time)
A Jumo 004 derived turbine and combustion section mated with a single-stage centrifugal compressor might have been in that vein.
On the other hand, Ohain actually managed to get the diameter of his engines down considerably with the HeS-8, and applying that compressor+axial diffusor arrangement on a larger scale might have worked quite well too. (the merits of the annular combustion chamber design in use is another matter though and -along with the uncooled turbine- one of the bigger problems with the engine) Perhaps adopting Muller's combustor and turbine designs or portions of them would have helped with this.
Regardless of this, the HeS-011 was a waste (costly and impractical design) ... and cancelling the 006 (HeS 30) was an even bigger mistake. And while Heinkel really stretched their design teams thin with all those jet and ducted fan designs, it still might have made sense to have 2 working designs in progress, perhaps with as much commonality in the hot section as possible.
However, on the note of the HeS-6. They had a working, flight-quality 1300 lb thrust engine in late 1939 yet they abandoned that in favor of Muler's axial design and sightly later began the HeS-8 design as a back-up. It seems that it would have been much more prudent to stick with what was already working and continue developing the HeS-6 to something practical for mass pruduction and service (even if ideally to be retired in favor of more advanced replacements fairly soon). It's a bulky design that would not have worked on the Me 262 or existing He 280, but given the He 280 was still on the drawing board at the time, adapting it to allow mid-wing mounting (or high wing with large under-slung nacelles) of the HeS-6 would have made plenty of sense.
At worst it would have made a usable, if short life and fuel hungry (existing HeS 3 and HeS 6 used 1.6 lb/lbf/hr -though some sources claim improvements on the HeS 6, the above pdf document states otherwise and appears to be a better authority on these engines than most others) engine at a time when nothing else was production ready ... or possibly proved unsuitable for service use but at very least allowed the He-280 to fly earlier. (important for flight testing AND especially for Udet's conditions for the Heinkel-Hirth merger hinging on the He-280's flight under jet power)
It was making thrust in 1939 that the HeS 8 didn't manage until around the time of its cancellation in 1942 for only modestly more weight, less complexity, and a good bit larger diameter. (appears to have been around 38 inches wide to the approximately 30 inches of the HeS 8 and 36 inches of the HeS 3 -from the drawings I've seen, the HeS 6 used a similar diameter impeller to the HeS 3, but much broader chord of the blades; this would allow for a more modest increase in diameter and also explain the higher mass flow but identical 2.8:1 compression ratio to the HeS 3 -larger diameter centrifugal compressors usually increase compression as well as mass flow)
At best, it might have matured into a practical longer term design and possibly even proved the radial turbine design had more merit than seen historically. (to the extent that it might have been worth retaining in scaled-up designs, but honestly, the bulk, weight, and use of metal resources makes it unattractive even if it could be practically air-cooled and more durrable than axial counterparts -BMW managed impressive turbine life trials by comparison)
--- One possible exception for this is if a plain, cheap, mild steel radial turbine was feasible, but this would still probably only be useful for short-life engines. (either single use or very regular turbine replacement)
As it was, with strategic metals not being stockpiled at all, producing less resource efficient engines earlier in the war wouldn't have compromised later productivity at all (but detracted from other machines needing those resources OR requiring additional procurement -Finnish Nickel in particular was bottlenecked by transport and not availability prior to very late war when that supply was blocked entirely)
And one other side note, and one that I don't recall often coming up on these sorts of discussions, but: Heinkel might have gained more support from the RLM if he'd put some resources into jet bomber/attack aircraft development early on. (maybe even to the extent of extracting resources from the conventional piston engine bombers then in development -especially the impractical and costly He-177)