There was no registration for 2 years, 1976-1977. So if you turned 18 in that time frame, you didn’t register. As you note, we returned to draft registration in 1978, which of course was under Carter.
Reagan actually campaigned on doing away with Selective Service as a waste of taxpayer money. All it does is pay armies of bureaucrats to compile a huge mailing list. If something goes down most of these people won’t be found, or if they are many will resist and clog the court system.
After getting elected he changed his mind as it was thought this would send the wrong message to the Evil Empire.
Registration was reinstated in 1980.
I know because I was caught in that net.