I mis-spoke above about the 23 mm cannon US Army project: one of the initial prototypes was a Browning Long Recoil weapon with similar roots to the M4/T9 37 mm weapon, and it was rejected mostly due to its weight still being above the specification. However, it used a different cartridge case and feed system (apparently a 50 round pan magazine) than the T9 and wouldn't be a simple barrel swap (and possibly recoil spring change), so the question there would be: as an interim measure, why wasn't a more conservative modification to an existing production weapon considered? (even with poor rate of fire and not much weight reduction over the M4/T9, it would have better ballistics ... though if necking down the existing 37 mm cartridge and using the same receiver/breech/recoil mechanism, 23 mm would probably be overkill, and something in the 25-30 mm range would have gotten ballistics more in the Oerlikon S/FF S, Hispano, or .50 BMG range while having better shell capacity)
I'd think a weapon scaled down further than the M4 would be more realistic for a cartridge closer to the specification the Army had in mind. (it still would've been a relatively mediocre performer, but smaller and lighter at least, possibly more in the realm of the 23 mm Madsen or Hispano)
Additionally, the Madsen guns (of all calibers) were definitely turned down primarily due to cost/complexity of manufacture specific to their mechanics. (the Browning Long Recoil system was a good deal more economical, if still more complex/costly and much slower firing than the Browning Short Recoil system of the M1919 and derivatives)
https://www.forgottenweapons.com/wp-content/uploads/2015/05/Aberdeen test of Madsen 7mm and 20mm.pdf
see pg 12 for concern over manufacturing complexity of the Madsen mechanism (in particular, they cited the use of many complex curved surfaces being used, requiring complex, skilled machining operations)
It's possible that some or most of those curved parts could've been made cost-effective as stamped or possibly forged parts, but I'm not sure when that became a viable option in US industry and it doesn't seem to have occurred to anyone at the time.
OTOH, if the ballistics of the 23 mm Madsen gun were considered at least acceptable, or if the US Navy had taken a greater interest in such (in spite of somewhat lower velocity), they also might have considered adapting the 20x110mm Hispano or 20x110RB cartridge of the Oerlikon Type S and FF S to 23 mm in a similar manner to the Madsen cartridge.
The Oerlikon gun might have been more attractive in this case as it would take a bottlenecked case and make it straight or nearly straight walled and thus get a fair bit more efficiency out of the same spring, receiver, and bolt of the existing FF S design: blowback and API blowback weapons favor straight-walled case as bolt-thrust has a nearly linear relationship between the ratio of the area of the base of the projectile and the base of the cartridge, which is also why the other 20 mm Oerlikon guns and German and Japanese 30 mm API Blowback weapons also use straight-walled cases. The chamber would still need to be strong enough to withstand the pressures involved, but that would be part of the barrel swap, and might otherwise mean using the FF S unchanged (albeit developing a belt-feed system for it would be high on the priorities and working on modifications to improve rate of fire), but I'd think the overall weight savings and performance compared to adapting the Hispano to 23 mm, even before they realized the reliability issues, would have been major considerations for the US Navy, plus the potential to switch between 20 and 23 mm configurations relatively easily and use the same ammunition as the Type S anti-aircraft guns being adopted on various warships. (this latter bit might also be a reason it would be a more likely consideration than the lighter, shorter, lower recoiling FF L cannon, which had similar muzzle velocity and roughly similar ballistics to the pre-war .50 cal loadings) Had they opted to copy the 174 gram Madsen projectiles and employ them in a modified FF S cannon, the muzzle velocity should've been around 770 m/s and sectional density somewhat higher than the 122 or 123 gram HE/HE-I/HE-T projectiles of the type S. (if they dropped to a rather light 162-ish gram projectile, it should've allowed around 820 m/s with similar sectional density to the 20 mm, but unless a relatively long, streamlined, but light projectile was developed, it would have a poorer BC and poorer ballistics than the 20 mm; a relatively short, light 23 mm shell with a fairly long, streamlined nose cap should have been easy enough to engineer if they were after higher velocity and good ballistics, plus ballistic shaping should be easier than with the typical 20 mm projectiles)
Blowback Gun Design note the section on bottlenecked case designs
The Hispano, OTOH would either have to reduce the power of its ammunition significantly to use the same mechanism, or strengthen the mechanism and add weight to cope with a more powerful cartridge. (the Madsen 20 mm cannon started out with an already somewhat more potent cartridge, so converting to 20 mm and adding a belt-feed mechanism didn't impact weight or rate of fire much)
Also, not related to US developments, but the pre-war, high power 20 mm autocannons of the IJA and Germany, both derived from anti-tank and AA guns, while rather mediocre and overweight to be efficient fighter weapons, used ammunition/cartridges that could/should have adapted well to 23-25 mm variants via simply necking them up. The MG c/30L and Ho-3 respectively firing the Swiss 20x138mmB and Japanese 20x125mm cartridges. The ground-based Flak-38 counterparts to the MG c/30L managed at least 450 RPM in later developments, so it seems reasonable that could've been maintained, and necking it out to 23 or 25 mm (possibly using Danish 23 mm or French 25 mm Madsen shell manufacturing to supplement such) should have been straightforward for producing a pretty potent early-war anti-bomber weapon, and hypothetically one that could also be synchronised and mounted in Fw 190 wing roots. (with a new cartridge, it should also have been possible to adapt it to the same 30 mm shells as the MK 101 was using for a low/medium velocity gun closer to MG FF ballistics and probably moderately higher velocity, but slower firing and somewhat heavier than the MK 108, but also available much earlier in the war and vastly lighter and faster firing than the MK 101)
The Ho 3 likewise should've used a necked-out cartridge relatively easily and possibly a 30 mm cartridge. (using the fairly light 253 g 30 mm shell of the later Ho 155 gun, velocity would've probably been closer to 600 m/s or slightly less, allowing for similar or slightly lower muzzle energy than the 20x125 cartridge and gun weight and rate of fire would also likely be similar to the Ho 155, but it should've been a much earlier war weapon) OTOH, 23-25 mm would seem a more obvious route and probably fit the existing cartridge case better. (I had a hard time finding the dimensions of the 20x125 Japanese cartridge ... the Swiss 20x138B has measurements listed online and wouldn't practically allow more than 25 mm)
Additionally, the British, during their trials of FN Browning aircraft heavy machine guns could/should have considered those chambered for the 13.2 mm Hotchkiss gun, or the 13.2x99mm derivative (basically .50 BMG necked up to take the 13.2 mm Hotchkiss projectiles) as they were loaded to higher velocities with better armor piercing characteristics than either American or British (Kynoch Export loads were slightly heavier and hotter than pre-war American .50 loads) cartridges and on top of the high rate of fire (1050 RPM in British Trials) those guns should've been seriously considered to fit the sorts of qualities the 20 mm Hispano was being developed for, but with the advantages of being easier to adapt to existing fighters as wing or synchronized nose/fuselage mounts. (ie Gloster Gladiators could even be re-armed with them, and the high ROF of FN guns should have made synchronized mounts much more reasonable than the 500-550 RPM typical of American M2 Brownings) The 13.2 mm projectiles also should've had marginally better incendiary capacity for what that was worth at the time, though would become more important once improved incendiary and API developments were introduced.
A lighter Browning derivative developed around the .50 Vickers cartridge should also have been possible, but would've been less ready-made and have thus required greater interest (plus further development of the Vickers .50 gun was more appealing for a time as it was initially more reliable and lighter than the .50 Browning in 1920s testing) but that cartridge had mediocre armor penetration and thus lack of interest. (I'm not sure why lighter, higher velocity AP bullet loads weren't developed, both for aircraft and AA use, as the existing ball and AP loads in the 2500-2540 fps range were rather mediocre, and while already light for .50 cal weapons at 580 grams, should still have seen similar advantages for lightening + higher velocity and higher muzzle energy on top of much better armor penetration within effective ranges ... plus a good deal of production had used flat-based projectiles, so lighter ones with better ballistic shapes could have had better all-around ballistics on top of good armor penetration, relatively light recoil, and relatively light gun weight) It's also noteworthy that the bullet shapes of .5 Vickers cartridges were already superior to those of their export derivatives used later by the Italians and Japanese and the 1920s era Vickers .5 inch aircraft gun was 24 kg compared to the 29 kg (and similar 700 RPM rate of fire) of the Breda-SAFAT heavy machine gun used by the Italians, which was perhaps the most mediocre one of the war. (the WWI vintage Vickers gun had also been prone to jams/stoppages in aircraft use, but I'm not sure how that translated to later marks or the .50 cal guns and their switch to disintegrating metal link belts, or the comparable .303 Vickers marks that had been lightened and sped up to 900 RPM that lost out to the still lighter and better performing .303 Browning, though the Japanese Army and Navy used similar 900 RPM vickers developments)
I imagine the license to manufacture .303 Browning machine guns would've been competing with any hypothetical interest in a 13.2 mm weapon. (and I'd think either would go through FN, since they generally dealt with Colt/browning gun licenses in Europe)