The US did actually develop an assault style weapon, the M2 Carbine, which was an automatic M1 with a 30 round clip.

It was too late for WW2 beyond some Marines in Okinawa, but in Korea, the US noted pretty much what you said, that inexperienced soldiers tended to panic and blow through their ammunition, while experienced ones proved highly effective in places where short-ranged engagements were likely.

The STG 44 had better range and power, of course, but it's also much heavier, to the point that the M2 might have been preferred if the US was given a choice between the two.