One must consider what is 'better' about smaller diameter faster cartridges. The purpose of the machine carbine is to dominate the gap between pistol arms length/pointy stick range and LMG fire range in a portable cheap small firearm. Say 25 metres to 200 metres. Beyond that you use an LMG and standard infantry rifle. If naughty chappie is over 200 metres away stop firing at him with a machine carbine as you will annoy him and draw his fire. Call upon your crew automatic weapon to make him stop it and go away.
The target remains the same and needs a certain amount of energy transferred to poke enough hole into it. To reduce weight you have to reduce energy. Nobody has seriously gone down the perfectly possible road of using .32ACP instead of 9x19mm and getting a lighter gun, lower recoil and more ammunition carriage on the soldier because it will not reliably stop the naughty enemy soldier nor poke a big enough hole in him. In a pistol the .32ACP lighter recoil allows the less experienced or smaller shooter to more easily handle his gun and place his rounds. Rather as the Enfield .38-200 revolver did when it replaced the .455" Webley which overwhelmed the inexperienced normal user. As a PDW at close I can see a small 32ACP machine carbine in that role but not as a weapon to dominate out to 200 metres.
We have 3 classes of weapons possible and 3 classes of cartridges, with some overlap.
We have the locked breech-guns, higher powered and expensive to built. (few, if any guns built?)
We have the delayed blow back guns, a bit lower powered and a bit cheaper.
We have the blow back guns which fall into two sub categories. The Pre-WW II gun maker style (still expensive) and the stamped (or tube) steel el cheapo's that showed up very quickly.
In the 1930s there was some argument over which direction to take. Including the Czech ZK-383
Which not only had the bi-pod but had a quick change barrel. Also had a detachable bolt weight to change the rate of fire, but you had to take the gun apart to do it.
There were several guns operating as delayed blowbacks, usually for the 9mm Mauser export cartridge. The delayed blow back guns generally had a slower rate of fire than the straight blowback guns. BTW the Thompson gun was supposed to be delayed blowback but they found they could leave the locking piece out with hardly a change in rate of fire and the bulk of the war time guns used the simpler blowback set up.
The French went the other way
using the 7.65 Longue and it was only long in comparison to the 7.63 APC. The gun only weighed about 6.3lbs despite being made of machined steel.
With on of the two lowest powered rounds used in submachine guns in WW II the light weight was not a problem.
Many of the 1930s (and earlier) guns had very optimistic sights.
Without getting into the area of intermediate rifle cartridges (7.9 x 33 mm German) the 9mm Mauser was about as powerful as guns got. Around 1360fps with a 128 grain bullet for about 530ft/lbs muzzle energy. (pistol ballistics)
Perhaps the US could have made a rimless .357 magnum for submachine gun use (9 X 33mm) with a straight case but it would only gain 100-200fps over the longer 9mm rounds already in use. And a blowback .357 mag gets a little scary.
as far as a "smaller" cartridge goes the late 50s (or 1960) they tried necking the .357 mag down to .257 (called the .256 Winchester Mag) and got 2200fp out of an 8 1/2in barrel using a 60 grain bullet. But you still need a .357 mag size action and you still need the long case so all you have lose is some bullet weight.
Note that this gets you about 200fpm more than the M 1 carbine with a bullet of almost 1/2 the weight, granted a shorter barrel on the Carbine will have some velocity loss.
These short barreled high pressure cartridges have more muzzle flash and muzzle blast than the standard 9mm rounds.
The 7.63 Mauser and 7.62 Torkarev were pretty much necked down 9mm Mauser.
I am not seeing any significant advantage to the smaller diameter bullets in a submachine gun. Not saying there isn't any, but adding 20-40 meters of range to a 150-200 meter gun may not have been worthwhile?