No, it wouldn't. All it would require is a system similar to motor vehicle licensing; with testing that includes safety, proficiency, legal understanding of use and responsibility, and cognitive ability. And yes, some background checking, too.This sounds lovely, but would require state intrusiveness and snitching to be the bedrocks of your society for such policies to be even mildly effective.
For example. If we denied a license to own, use, or carry firearms to people who have been convicted of DUI, we could save many hundreds of lives every year from gunshot. Same goes for people who have been involved in stalking, domestic violence or abuse, making threats, reckless driving, and so on. Because we already know these people are not stable, and are not socially responsible. We don't need anyone to "tattle" on them. We just need to keep track of the information we already have, and apply it sensibly.
Selfishness is not freedom. We live TOGETHER in a modern, highly interconnected society. And that means we don't get to do whatever we want just because we want to. It means we have to consider the effect of our behavior on those around us. And since some of us won't do that, it means we have to establish systems of governance that will do that for us, and see to it that we follow through on that responsibility. It's why societies form governments and pass laws to regulate people's behaviors in the first place. I'm sorry that you don't like that, but you seem quite happy to avail yourself of all the amazing benefits that come from living in a modern interconnected society. So now you're going to have to put on your 'big boy' pants and accept that living in such a society comes with a responsibility to the well-being of all those people you live in this society, with. And might mean that you don't get to play with guns the way you'd like to.What you want is your restrictions implemented. What you're unwilling to consider is that doing so requires the subjugation of individual wills. And the best arguments you have to convince people to bow to your agenda are emotionalism.