First a note to all people, LEO's had FA's even BEFORE the 34' act.
Yes, and the average American could order a Thompson Submachinegun, the MP5 of the time from a Sears catalog and have it delivered anywhere with no documentation, background check etc.
So if police lost one then it would not be a very big deal unless a child found it. It would be about as serious as misplacing a container of dangerous cleaning chemicals, could pose a danger to a child, but otherwise not a huge concern.
Police could have them, and all citizens could have them. There was not even any prohibitions to felons and previous criminals from having them once they rejoined society until 1968.
So no matter who found it, it really was no big deal unless it was a child.
It is a huge deal to people now because of all the gun control. We have acceptable classes of people for some things, and less than equal for others. When the sheep might have something only allowed the shepherd people panic.
Prior to the NFA act some police did use firearms like the thompson. Most did not.
After the '34 act most LEO did not have full auto weapons or deploy them.
LEO were considered little different from anyone else and subject to most of the same laws until much later.
Even when SWAT teams first started in the LAPD in the late 60's (after seeing the SAS perform in the wonderful UK.)
Many of those weapons had the same appearance as full auto and the public assumed they were. They were in fact semi auto firearms. However after the public assumed for so long they were full auto, and it become generaly acceptable that they were, they actualy did switch over to select fire weapons. They of course used a current incident as in all situations to add that additional justification. The incident with the Symbionese Liberation Army, however similar incidents had happened previously, and police had faced criminals armed with such weapons for decades, so it was simply a convenient excuse.
Here is an section from wiki, which I know to be correct from other sources long before wiki existed:
The normal complement of weapons was a sniper rifle (apparently a .243-caliber bolt-action, judging from the ordnance expended by officers at the shootout), two .223-caliber semi-automatic rifles, and two shotguns. SWAT officers also carried their service revolvers in shoulder holsters. The normal gear issued them included a first aid kit, gloves, and a gas mask. In fact it was a change just to have police armed with semi-automatic rifles, at a time when officers were usually issued six-shot revolvers and shotguns.
As you can see most police did not have full auto weapons after the NFA act, and in fact it was frowned on when they had more than revolvers and shotguns. LAPD was on the "cutting edge" in changing that trend and it did not even equip officers with rifles until the late 60's. No full auto/select fire until the mid 70's.
In the interim various laws in various localities and states began to appear that excluded LEO. Prior they were generaly considered equal to everyone else. As more and more new restrictive laws appeared they excluded law enforcement.
The result is what you see now.
Prior to the political climate of the 70s the average American would have been appalled that the police were militarizing. The federal military was specificly prohibited from being used against the citizens since the Posse Comitatus Act, and turning the police into paramilitary branches would just be a way to bypass that. However the disdain by many for the radicals of the time caused the citizens to allow severely restrictive arms laws, where they could be carried, how they could be transported while at the same time giving more special privelidges to LEO than ever before.
Most current gun control stems from then. Prior to that it was legal to walk down the street many places with a longarm, take them into federal buildings, possess them while picketing, carry all firearms openly most places (and even concealed in many) etc.
