Didn't Reagan also sign the so called Firearms Owners' Protection Act of 1986 which closed the NFA registry for civilian owned machine guns? Sorry he was not the pure defender of the bill of rights many here think he was.
That prohibition was a last-minute voice-vote amendment to the bill. The problem with large bills is that stupid little things can be snuck in and signed off without being noticed. Nobody knows if RR knew the MG ban had been inserted.