I agree that screaming "my rights" isn't going to do anything but preach to the choir. I'm not suggesting that we do that. Nor do I think we have a system of absolutes, with Team: Guns pitted only against Team: No Guns. That's the apathetic middle I mentioned. While most of the country, if polls on the matter are anything to go by, seems to fall in the framework you've suggested, they're also wholly uninformed on the matter and generally unwilling to get involved in the details. They'll look at the TV when another shooting happens, say "oh my God, that's terrible, we have to do something, this is an epidemic," accept whatever cookie-cutter solution is presented as effective, and go back to doing whatever it was they were doing before. That's illustrated in the polls after Sandy Hook vs. those conducted today. The apathetic crowd is the reason why we're in a state where I'm one bad shooting away from being limited to ten-round mags again.
The fight for their hearts and minds is important, no doubt, but we're in a less-favorable position than our opponents. We're not seen as the "stop violence" position, but rather the "stop messing with my stuff" crowd. The two aren't mutually exclusive, of course, but rarely do we offer actual solutions. And when we do, those solutions generally involve "more guns," a position which is easily caricatured by the other side. Frankly, ours is a difficult position to articulate to a group of people who scream "we've got to do something" every time something bad happens.
The surface validity of UBCs is addressed in my first point on the matter. They're completely unenforceable without a method of verifying who has possession of what. That system isn't currently in place in most states (California has a method of doing so, some sort of safety certificate IIRC). And while on the surface I'm fine with the idea of having everybody go through a check -- actually, I'd like to see it -- I don't like the means which would be required to institute such a thing. That's my objection to UBCs in a nutshell. You can pass all the laws you'd like on the matter, but without registration or something like it, all you're doing is creating meaningless bureaucracy. I don't think that's an argument that the middle wants to hear. To them, it's just another roadblock to "do something."
You pointed out earlier that many of our high-profile mass shooters wouldn't have been stopped by our current checks, and that post-Virginia Tech we've attempted to strengthen reporting on mental health issues. Again, I'm fine with the idea of increasing reporting. My question is: how do you catch people who aren't currently in the system, as most of the shooters were not? On the surface, the "gun violence restraining order" idea seems like it would do that. My problem there is that the current California law seems ripe for abuse, and it reverses the adversarial process currently in use in our justice system (I have to prove that I'm not a danger at the hearing, rather than the other side having a high burden of proof to meet to show that I'm a danger). Once again, perception as just another roadblock to the "do something" crowd.