> but you're always allowed to own a gun if you own your own home
That's not true everywhere. No state forbids ownership (though it's certainly more difficult in some states than in others) but certain municipalities deny ownership to many or most who seek it. Chicago did for many years until its laws were ruled unconstitutional. Same with Wahington, D.C., which changed its laws after losing in court.
In New York City it's still very difficult to own a weapon even just to keep at one's residence. It has high application fees and applicants can be denied for seemingly arbitrary reasons.
So there's no place where it's automatically forbidden for everyone, but it's certainly not true that a person is always allowed to everywhere in the US.