Reagan's words were as Republican Governor of California in 1967. He was definitely in favor of a Republican-sponsored bill repealing gun rights.
The problem was that the framers of the U.S. constitution thought of guns not as an item that all people should have, not especially for hunting and protecting their homes/family/property from just anyone, although that would be their primary use. The constitution only addressed the right for all citizens to carry guns in order to form militias in the event that tyranny reared its ugly head, within US Government agencies.
But the real problem is that there really were forms of tyranny that had been rearing their head in America for many years against blacks, American "Indians" and poor whites who could not pay their debts. The most violent tyranny was against the native American "Indians" but the most insidious was against blacks. And then after the constitution allowed more than just land-owners to become citizens they allowed blacks to become citizens, while they were still being tyrannized by tyrants in their own government.
That created a problem for the hypocrites running the US Government. A group of black citizens began watching some of the most tyrannical agents of the US government, the white-sponsored police in economically abandoned urban centers. What they were doing was called "copwatchting." But they were watching while armed with guns and, what's even worse for some, cameras. See Mulford Act in Wikipedia, for example.
Organized copwatching groups emerged as early as the 1960s in urban areas in the United States when the Black Panthers famously patrolled city streets with firearms and cameras, and other civil rights organizations conducted unarmed patrols in groups.
Obviously, Reagan, a rather dullard hypocrite, realized he didn't really believe in the constitution. In this situation it was easy to get both Democratic and Republican support. (Especially "Dixie Democrats" [in the Southern United States], most of whom would later become Republicans as soon as Northern Democrats began to associate the "Democratic" agenda with civil rights toward blacks.)