If the public is demanding it then the public should pay for it. Background checks are a law enforcement mechanism so taxpayers should foot the bill for them.
It's a fundamental miscarriage of justice to punish someone for being the victim of a crime. Indefensible in my mind. A crime against losing something is about equally egregious. In what other areas do we have these laws? Would you be willing to punish government employees and business owners under these same circumstances? LEO's lose weapons and have them stolen probably more often than we know.
Here's something else to keep in mind with a registry. Everyone dies. Then what? At least two of the firearms I was to inherit from my dad ended up going missing. Of the dozens I currently own, I give zero fucks if I'm dead or dying and the next owner isn't tracked. Why's that, you ask? As was pointed out by
@Gregolian, letting the government know where all the weapons are just enables tyranny via confiscation. And before you start pontificating how that's not a concern, here's what's going on in Hawaii. We have a registry. Not long ago all of us law-abiding folks got put on the federal rap-back list (or whatever that informs one state if you were convicted of something in a another state). More recently the Honolulu PD sent out letter to folks on the registry, who were also on the medical marijuana list, saying they must relinquish possession of their firearms. Although public sentiment squelched that shit quick, the specter looms. Now there's a bill in committee looking to essentially ban semi-autos, with no provision for grandfathering. The registry is about confiscation, not crime-fighting. I once asked
@A.C. how the registry helped solve crimes. His response (the specifics I forget) did nothing to make me think otherwise. Now add all that up, plus all the talk of disarming people with "mental issues" and there's my problem. The bar to be separated from you firearms will only get lower and lower. So even if the weapons are able to be purchased, the right and ability to engage in effective self-defense will only be available to fewer and fewer people.