All of the possible outcomes you listed are plausible. People will find ways around any "registration" requirements. Canada experimented with this method in 1995. In the end, the registration was ineffective and was repealed.

Modify behavior? Yep, that's the way to go. Licensing firearm users, with background checks and compentency check, will help a lot:

https://medium.com/common-sense-please/a-common-sense-approach-to-firearm-legislation-99899a2d017f

The police can confiscate any firearm that is in pocession of an unlicensed user. That will change behavior.

Even this approach will not address all of your possible outcomes. But it should cut them by half. That reduction is worth the hassle of licensing users. I think most gun owners would agree.