Anecdote: I know many gun owners who are fine with universal background checks if they can be performed at home for free in a reasonable amount of time. I don't understand why people who want gun regulation aren't pushing for this. Free, easy, and fast background checks.

Unenforceable without a registry of some sort. You'd never know if a check was ran for that firearm or not.

And indeed, it's when you mention the fact that UBCs require a gun registry to be enforceable that people really start opposing them. (The current background check system was passed with the compromise that it wouldn't be used to make a gun registry though the ATF, and in fact the ATF was not allowed to digitize records for exactly this reason. The ATF appears to have begun ignoring this deal in recent years, though)

The ATF keeps the GAO quite busy. It's too bad the GAO has no real power on this and many other issues.

This is a solved problem with modern cryptography. Governments just don't want to implement a cryptographic scheme like this, given the side-benefits of a registry for future confiscation, red-flagging, etc.