Sen. Rand Paul (R-Ky.) spent 60 seconds on a radio show explaining how everybody should have the right to enter into private marriage contracts, and the government should stay out of it. Then he spent five seconds explaining that Kentucky clerk Kim Davis (she who is refusing to hand out marriage licenses because of her religious objections to same-sex marriage) is "making a stand" by doing something that is "part of the American way" (listen to the response here).
Guess which part is ending up in headlines? It's his vague defense of Davis, obviously. To be fair, it is part of the American way for government functionaries to just not do their jobs, but that's not what he means. The problem here is that Paul is trying to stick to his consistent talking point—get the government out of marriage—in a situation where the argument doesn't quite apply.
He brings up the Alabama example in his response as an alternative to Kentucky's current system. Alabama legislators have proposed eliminating any responsibility for probate judges to "issue" marriage licenses altogether. Instead couples will fill out and turn in forms stating they are married (signed by two witnesses) for recordkeeping. Judges or clerks would have nothing to do with either solemnizing or having to care one way or another about the sexes of the people on the forms. They're just there to file the marriage forms away, and if there are any problems that require government intervention, it would be up to the courts to determine the validity of the marriage based on current law.
This is how Alabama is getting around conservative judges who don't want to support same-sex marriages (and state law gives probate judges the ability to decline to issue marriage licenses altogether).The important thing to understand, though, is that this system would be in complete compliance with the Supreme Court's Obergefell v. Hodges ruling. The state will recognize same-sex marriages under the law. And it's not going to stop same-sex couples from getting legally married. It's just removing itself from playing a role in the marriage process.
That's a little different from what's going on in Kentucky. Davis' refusal to hand out licenses prohibits couples who go to her office from having their marriages recognized by law. It's almost pretty much the opposite of what Alabama has done.
There's also North Carolina's solution, where clerks can refuse to hand out marriage licenses due to religious objections, but it would be up to the county courts to make sure somebody else will pick up the slack. It's an imperfect solution that could end up being expensive, but it's an attempt to let everybody get what they want.
These aren't options actually being presented here, so Paul's response ends up feeling like a digression rather than an answer. The defense of Davis more directly responds to the debate non-libertarians are having in the media, so that's the part that ends up in headlines.