>would a $1000 icepick machine with $50 20 round magazines do better than an ultra cheap rifle using a round that is always ballistically effective on a person out to 200 yards and still viable at 250-300
No. Comparing FMJ to FMJ, the only permanent advantages 5.56 has are maximum range and accuracy, but the M1 Carbine was never meant to be a long range weapon, and the .30 Carbine's greater surface area and momentum mean more reliably effective terminal ballistics. It doesn't depend on super high velocity and rotation to wound, it does better just by hitting. Remember, we'd have to use the bullet technology available in WWII. No cop killer armour piercing exploding meme NATO ammo.
The 5.56 does have more kinetic energy and that seems like a huge boon until you remember how light those bullets are and how poorly they perform through brush, debris, and concealment, which any WWII battlefield is full of. Regular FMJ does have a tiny chance of fragmenting, but the rifle has to fit into the profile of the M1's role - that means a barrel that is short enough to risk effectiveness (if we want equivalent size to the M1, which we have to for paratrooper and vehicle use, that means the Mini-14 is using a 16" barrel, where 5.56 FMJ likes 20").
Basically, it would be a fuck of a lot more expensive, offer questionable ballistics, give no advantage in firepower, and would be worse in many situations.