If they did, then their armour-piercing capability wouldn’t be as good as it might seem on first glance.
The only projectiles that the 37-mm M1916 and M1918 used, the American derivative of the 37 mle 1916 T.R.P. and the 37 S.A. mle 1918, respectively; were the Mk. I and Mk. II high-explosive which could operate as a semi-armour-piercing projectile. Due to that, no armour-piercing projectile was ever developed for the U.S. 37x94R cartridge.
For the French cartridge, the best that would be available would be the 37 mle 1897 projectile, which was previously removed from the game post-penetration calculator due to its penetration making it unusable. Inserting its statistics with the velocity of the M1924 guns, you get 15 mm of penetration at 1,350 fps (411 m/s) and 27 mm of penetration at 2,000 fps (609.6 m/s), which would make only the latter at all usable.
Base-fuzed high-explosive has its own modifiers in the penetration calculator that I don’t know, so I can’t make an estimation of how the U.S. cartridge would perform under those velocities.
Though, looking at some more stuff, gun development seems to have followed this trend:
- 37-mm M1921, sometimes misidentified as M1924 and would have been the T1 but never retroactively received that designation; Browning developed it by enlarging the cal. .50 M1921 (predecessor to cal. .50 M2) firing the normal 37x94R cartridge.
- 37-mm M1924 No. 1, developed in 1924 to meet the 2,000 fps milestone and used an enlarged 37x120R cartridge, failed to exceed 1,350 fps.
- 37-mm M1924 No. 2, later renamed to T3 in 1926 when new nomenclature system was introduced which is also the same T# as what was supposed to be the first prototype of the 37-mm M3, though I can’t verify if they’re actually the same guns; developed in 1925 and used the further enlarged 37x123B cartridge, allowing it to meet the 2,000 fps requirements for an aircraft gun.
- 37-mm M1925 No. 1, developed in 1925 and used a further enlarged 37x123B cartridge but failed to meet the 3,000 fps requirement for an anti-aircraft gun.
- 37-mm M1925 No. 2, later renamed to T2 the same year when new nomenclature system was introduced; was the M1925 rebuilt in 1926 to use a new 37x224R cartridge which allowed it to meet the 3,000 fps requirements.
- 37-mm T2E1, a variant of the M1925/T2 developed by Colt which thickened the barrel and used a new 37x223R cartridge modified from the 37x224R cartridge.
- 37-mm M1927, T2E1 is standardized for mass-production in 1927, and then immediately after redesignated the 37-mm M1 anti-aircraft gun, only three built
- 37-mm M1E1, M1 guns modified in 1937 to reduce the muzzle velocity from 3,000 fps to 2,800 fps to reduce wear and improve accuracy
- 37-mm M1A1, M1E1 guns standardized in 1938 after the removal of the water jacket, still not mass-produced
- 37-mm M1A2, M1E1 guns further modified in 1939 to reduce their muzzle velocity from 2,800 fps to 2,600 fps, eventually over 7,000 produced during the Second World War
- 37-mm T9; was an eventual derivative of the T2 for use as an anti-aircraft gun, was then standardized to the M4 and used a 37x145R cartridge, famously equipped on the P-39 “Airacobras” and P-63 “Kingcobras.”
As far as I can tell, the T1E2 and the T1E3 used the 37-mm T2 with the higher-velocity 37x123B cartridge to achieve a velocity of 2,000 fps (609.6 m/s). Meanwhile, the T1E4, the T1E5, and the T1E6 all used the 37-mm T3 with the lower-velocity 37x120R cartridge to achieve a velocity of 1,350 fps (411 m/s). The reason for the switch was probably due to what I described above, with the T2 wearing down the barrel far too quickly due to how thin they made it. Until the later 1930s, the armour-piercing capability of the First World War-era 37-mm M1918, and by virtue of the lower-velocity 37-mm T3, were still acceptable.
The T2 Medium also used a Browning and from photographs of it, it seems like it used the higher-velocity 37-mm T2, too. This was removed and replaced by a 37-mm M1918, which itself was later replaced by simply a cal. .30 M1919, around the same time as the switch was made from the T2 to the T3 on the T1 Lights.
Though, through all of this, I still have no idea what their actual performance is beyond their muzzle velocity. They don’t use enlarged Hotchkiss projectiles, I just got confused when looking over stuff initially in reading the 37-mm M1921 as having used an enlarged one. Similarly, the suggestions for these tanks made by @CaID on the old forum seems to be inaccurate, listing the guns as using the 37-mm M80 API-T projectile. However, this is impossible since that projectile wasn’t developed until sometime after the 37-mm M4 (T9) was introduced in 1942, long after all testing had ceased on both the T1 Lights and the T2 Medium, and the projectile itself is chambered in the 37x145R cartridge, a cartridge that neither the T2 or T3 guns were ever adapted for which only used either the 37x120R, the 37x123B, or the 37x224R cartridges.
The one other notable thing I found is that apparently in 1935 a T2 gun was tested with the 37x224R cartridge (3,000 fps) as a tank gun. However, I don’t know if they would have actually tested it on any of the tanks. I know that the T2 Medium was not an option, as work had already been halted on that in favour of Christie’s T3 Medium and that the 37-mm T2 it had was already removed by 1931, replaced by the prior-mentioned 37-mm M1918 and then the cal. .30 M1919. The T1E4, the T1E5, and the T1E6 were all equipped with the 37-mm T3 that could only fire the 37x120B cartridge. By that time, the T1E2 had already been rearmed with the 37-mm M1918. That leaves only the T1E1s as a potential option for having been tested with the 3,000 fps cartridge, but I have no idea what their status were by 1935.