I've talked about this Catch-22 in this thread before. Gun rights activists oppose a lot of reasonable measures out of fear that they will lead to bans and confiscation, but that very opposition is a large part of why the other side of the argument (for which I can't think of an term for that doesn't implicitly criticize either side) has a tendency to jump straight to bans as the first response to an incident.
It doesn't help that there is a lot of misinformation out there, and there are more than a few "god in the gaps" cases where mockery of one side or the other was mistaken (generally in good faith) as that side's actual arguments.
One quibble - gun licenses are not universal in the US. There is no Federal provision for them, although many states require one. Personally, I think going to a national licence would be a good thing, as I've advocated before, but there would have to be some sort of reassurance provisions to ensure that this isn't really a slippery slope.
Background checks are nearly universal, and there's plenty of support for making them universally so, but that isn't quite the same thing.
I think you need to rethink some of the terms you pull out. What you're calling catch-22 there should probably be called a "viscious cycle", and where you say "god of the gaps" you really need "Poe's Law" or something.
"God of the gaps" isn't really fitting there, at all. A "god of the gaps" argument is one in which you use your pet theory to explain some phenomena, but as alternate explanations are found for parts of it, you say "well all the bits you haven't explained yet - that's totally explained by my original theory - so I was right all along". It's related to the concept of "shifting the goalposts" too.
One area you might see a "god of the gaps" argument in a secular context is in e.g. a nature vs nurture debate. e.g. if someone is a die-hard behavioralist then their starting premise is that 100% of human nature is explained by nurture. But then, you explain
one thing to have a biology basis, and they go "ok, i'll accept that in
the most limit context possible, but everything else ... completely explained by my original premise". e.g. if you prove that, for example, 10% of variation in sexual orientation is explainable by the fraternal birth order effect (which has strong evidence that it's biological and not social) then a die-hard behavioralist using a "god of the gaps" argument, will turn around and say "ok, biology explains
10% of sexual orientation - but the remaining 90% is clearly social!" The flaw here is that if
10% of something is proven to be caused by
one biological cause, then it becomes
more likely that additional biological causes will be discovered which chip away at the 90% remaining.
Basically why this is a flawed way to debate is because the person is putting an undue onus of proof on the other side, while absolving themselves of needing to provide evidence. e.g. if you provide evidence that say, 50% of human nature is explainable by biology, then it
doesn't logically follow that the remaining 50% "must" be explainable by environment - because that is the "god of the gaps" argument: the person is just
assuming their preferred explanation is correct for all cases which lack evidence. But nothing makes
my preferred theory better than
your preferred theory, if we both lack evidence, which is why it's a fallacious argument.