The point is that it ultimately falls upon a citizen to prove that they are indeed legal. Most if not all states have a law that a person must present a drivers license if it is demanded by a police officer. I know your argument is that an officer can check it in the computer, but what happens if the computer is down? An officer now cannot check if you indeed have a license, but the law says you must have a license on you, guess who gets a ticket.
In Nevada, NRS 202.350 states in part: or expose for sale, or give, lend, possess or use a machine gun or a silencer, unless authorized by federal law. Short barreled weapons have a similar clause.
Now back to my point, without a system for an officer to check, it is up to the person to show they have permission by the feds. If you get caught with marijuana, do you really think an officer is going to assume you have a medical permit for it? It is a law of numbers; NFA weapons are rare when compared to the numerous illegal weapons out there.
I pull over alot of cars at work, but I rarely give people any tickets. I am a supervisor and have other things to worry about, but feel a duty to at least stop an offender for an obvious violation in front of a marked police vehicle. If the person cooperates and has no warrants, a clean record and a good attitude, they usually get a warning. But with a NFA weapon like yours, there is a ton of liability for an officer. If that weapon was used in a mass shooting and it was learned that it had been examined by the police earlier and it could not proven it was legal at the time and released anyways, that officers career and financial future are over.
Owning an NFA is a privilege, not a right. It is up to you to prove you are allowed to have it.