A background check is as good as registration. Why else would you need one?
I think they should have it like drivers licenses. If you want a gun you have to take a class on safety, storage and the laws for your area. You would then be issued a firearms license.
When you go to a gun dealer he would then run your license to see if it is clean. Having the license would not tie you to a particular gun just like you can have a drivers license without ever owning a car.
The license would also be your copncealed carry permit. All states should honor each others weapons licenses.
My point was,the Constitution does not say any such thing. In fact it says just the opposite. “Shall not be infringed” means it is none of the government’s business whether you or I or anyone has arms or not.