Seems like everyone and their dog managed to sneak back a service weapon. Own great-grandfather came out with three carbines. I'm aware that the US allowed/turned a blind eye to taking captured Axis firearms as war trophies, but how were so many soldiers able to successfully steal their weapons?
Also, what changed in US military policy and inventory control that made it significantly harder for American soldiers who fought in Korea and Vietnam to run off with an AK, M14 or M16?
In terms of policy and inventory control....
The US National Firearm Act (NFA) of 1934 put restrictions on ownership of fully automatic weapons in civilian hands.
Both the M1 Garand and M1 Carbine were semi-automatic weapons and as such were not affected by the NFA's provisions on full-auto weapons.
Many German weapons taken home were also either bolt-action or semiautomatic.
The M2 carbine (full-auto M1 carbine variant), BAR (Browning Automatic Rifle), Thompson SMG, M3 Grease Gun, and similar guns were all fully automatic and hence subject to NFA restrictions.
German/Axis full-auto weapons like the MP series, StG-44, MG34 / MG42, would have been covered by the NFA when shipped home.
IIRC in the 60's or 70's there was an amnesty for unlicensed full auto weapons, and the presumed target was war bring-backs.