Hi there, I actually answered a very similar question a while back that explains how M1 Submachineguns were distributed, and by whom.

The short answer is, for submachineguns, it was heavily dependent on formation, as they were considered vehicular weapons. They were company level weapons in standard Rifle units and were nominally handed out on a mission-by-mission basis (I go into depth in my answer on why that was often not the case). So your assumption that they were normally carried by NCOs is not entirely true.

Like most modern militaries with SAWs, whom got the BAR, on paper, was largely dependent on seniority and skill with the weapon - but in reality often translated to who could carry and fire the weapon for extended periods of time. Curiously, US Army Rangers, apart from other things that distinguished their training from standard units, were all required to qualify with the B.A.R. If you want to get technical, being able to carry heavier SAWs like the MG42 or BAR, and have the stamina to fire them, usually is all the 'skill' that was required. Finally, US Army training was understandably rushed for the majority of WWII due to manpower crises and rapid expansion; many machinegun platoons shipped without technically passing their qualification. My point here is; no, your qualification probably had alot less to do with who got the SAW than the test of battle and physique did.

A thing to keep in mind with US production values in WWII; with notable exceptions (such as the M1 Rifle) the US weren't just producing for themselves. Many M1928 and later M1 series submachineguns were handed out to the commonwealth nations in lend-lease stipulations.