On a time line the 30M1 would have been the first by one year. The German MP-43, MP-44,SVG-44 were behind the M1 Carb. 1942 by one year. My vote for the first Assault rifle was the MP-43 & 44 Select fire 500 RPM chambered in 7.92 X 33 Ctg. The Sturmgwehre was a true light machine gun for issue to infantry.
The idea behind the M1 30 Carb. was a replacement for handguns in "Panic" shooting defensive use. The Nazi MP-44 was designed for attack by fast moving offensive or assault units.
The Russian liked it so much they copied it and called it the AK-47. The Russian also took home all the Germans in door flushable toilets. They had no running water. The Communist were not idea people.