It wasn't until the end of the war with the advent of the Stg-44 did the concept of a mid-powered cartridge exist. Throughout almost the entirety of the war, there were only two infantry weapons - small submachine gun (Thompsons, MP4s) and high-powered battle rifles (M1 Garand, Mosin Nagant).
There was pretty much no concept of IBA during the war, so what drove the decision to arm infantry with weapons that were so powerful, insanely so in the case of the M1 Garand firing a .30-06 round, when that also meant carrying very heavy ammo, dealing with high amounts of recoil, and heavier weapons and barrels to deal with the pressures involved?
The trend with infantry main armament have been to reduce the calibre but increase the range and accurace for a long time.
16th century matchlock muskets were often .75 or even larger. 18th century flintlock muskets were often in the range of .63-.78. Later rifled perucssion cap muskets were often around .45-.58. These calibres were retained with breech-loaded rifles. With black powder, you simply needed a lot of gunpowder to send the ball any distance, and that required a large calibre.
Generally, the rifled musket allowed men to fire and hit at a longer distance - having superior range was important at least up to ww1.
The invention of the minie ball and later smokeless powder allowed the rifles to have a much smaller calibre - 6,5mm to 8mm were the most common. Post-ww2, armies realised that combat distances were not as long as their primary rifle rounds had been designed for, and most started adapting shorter and then smaller calibre cartridges that could do the job with less cost and less weight to carry.
So the insanely powerful cartridges of ww1 and ww2 were far less in calibre (but had longer range and much better accuracy) than earlier rifles. It was a natural result of the evolution of guns and ammunition.
Small correction that I hope is acceptable: the 7.92x33mm Kurz cartridge used in the StG 44 was designed before WWII, as was the .30 Carbine, so the concept of a mid-powered cartridge certainly existed before the end stage of the war. The M1 Carbine entered service in 1942.
Because on paper, a 1000 yard effective range for all of your infantrymen looks great. No one would have guessed the efficacy of an intermediate cartridge without first hand experience and study of the ww2 battlefield.
The caliber and load of WW2 ammo was better able to reach out and accurately hit targets +500m away. It also enabled the rounds to go through brush, grass and branches and still have stopping power.