:::: You can argue that the first "assault rifle" appeared in California in recent decades, because it's the first time there was a legal definition of what "assault rifle" actually meant. Otherwise, just what do we mean by the term?  Invented? In service? In bulk service?  With the performance of a military rifle? As there aren't many of such (really just the AK47 / AKM in full calibre, and that's a short intermediate cartridge) as it's hard to control, most are about .23 calibre rather than .303. But how small can it go and still be a rifle, rather than a sub-machine gun? 
 :::: The Avtomat has the problem that it's using a pretty useless cartridge, giving it the performance of a machine pistol. Nor can it use anything more powerful, owing to the overheating problem. Mostly though, it's just the limited numbers made - 100× as many Sturmgewehr than Avtomat. 
 :::: I would disagree Thomas' point that the Avtomat cartridge is too large, as the 6.5 Arisaka cartridge is so low powered, even though it's a large case.