I could see banning ownership of guns for current high school students, but personally I think the 21 YO drinking age is a bit silly, and if you can join the military, drive a car, or smoke at 18 then I don't think it's unreasonable to say that you're, theoretically, responsible enough to own a firearm. Not everyone is, but then not every 50 year old is, either, which leads into the next topic:
At the moment, I personally think this is the best route possible when it comes to gun use/ownership. You need a licence to operate a car, why not a gun? In order to drive a car, you have to demonstrate that you can safely operate one, so it makes sense to me to have to prove that you can safely operate a firearm before you're allowed to use one. As you say, a licensing system would allow relatively free ownership of firearms, just like it does with cars, though I do think a mental health screening/psychiatric evaluation should be a key part of obtaining a licence. I also like the idea of various grades, and requiring upkeep just like a driver's licence. As you say, it would make verification easier, so rather than a federal background check, you could simply check the licence and theoretically the background check should already have been done when the user obtained/updated the licence along with checks for any watchlists or medical issues. Theoretically I'd like to see a test that verifies that users A) are medically cleared to operate firearms (government subsidies for this might be helpful to keep access as universal as possible); B) have the theoretical (written) knowledge to operate firearms, deal with dangerous or unusual situations, deal with malfunctions, and work with law enforcement correctly in the event of an incident of any level of severity; and C) operate a firearm safely in a practical test including safe operating procedures, operation of safety devices such as safes or locks, the basic ability to hit a target reasonably well, and the ability to field strip a firearm.
For me, this view comes from the idea that gun ownership is a privilege, not a right. Guns and shooting can be a fun hobby or useful tools, but just like cars they're dangerous and aren't, in my opinion, essential for daily life. If gun ownership is an inalienable right like food or shelter, can you realistically draw the line and say that criminals aren't allowed to own them? I'm not sure you can, easily. But if you treat it as a privilege that you have to earn then it makes more sense to me.