Eh, you stop at things without substantive extra use, or whatever the word is. Can get by without a gun pretty easily, less so without a car in the US or being able to cross bridges. There's also this thing where firearm ownership is one of the jagoff huge statistical gribblies when it comes to suicide, heh.
... any case, I'm pretty sure we actually do ban people with certain mental illnesses from driving cars. Not like there's not stuff with perceptual buggery that renders it impossible to pass the requirements for acquiring or maintaining a license.
Firearm ownership is a right, not a privilege. The distinction is important.
If you're gating ownership with a psych check, it's a privilege, heh.
Legal question in the US is kinda' sketchy, too, though. Most precedent has came down in favor of individual ownership, but there's still a bit of an argument going on on the subject. It's damn sure not the same sort of right as stuff enumerated in the first, in any case.
People are (rightly!) concerned about a list like that being created for the same reason EnigmaticHat is concerned above about the creation of an active database of people with mental health problems. It cannot possibly exist for any purpose other than to discriminate, legally or otherwise, against the people whose names are on it.
It totally could exist for other purposes, really. Buncha' sociological shit tracking demographics and ownership patterns and whatnot could be done without even knowing specific names. The legal/enforcement possibilities are pretty obvious, and don't preclude things besides discrimination (one of the common talking points is that a database would make it a lot more certain whether a firearm used in a crime
isn't yours, if you've been accused, ferex), either. Active stat tracking is useful stuff even if you're not doing much particularly heavy handed with it.
Potential abuse is obvious, of course, but the question folks ask are if gun crime and the potential issues caused by mental illness is worth the potential. Note that I ain't saying they
are worth that potential, mind yeh. Just that saying the stuff would be used for discrimination isn't exactly a counter to the arguments for it, given that even with the quibbles above, to a fair extent that's largely the point.
E: Though, all that said, as RK noted while I was typing, as a country we're apparently pretty okay with stripping people of rights a metric fuckton more fundamental than the ability to own a specific tool for killing things. Particularly so far as how the GOP approaches the issue, it's a farce to claim all that many folks really give a shit about whether it's a right or not, or would care even if it was.