Let me just start out by saying that in the particular case of the topic of free will; in particular, for the philosophical consideration of free will, our intuitions about what free will is are of major interest to philosophers. This is why PAP was such a big deal for libertarians, why Frankfurt cases are such a major concern, and why the entire project of experimental philosophy began in the first place.

The main reason I would offer that we focus so much on intuition is because our understanding of free will is based more on reference than definition. That's not to say that our intuitions are necessarily valid; it's entirely feasible that, for instance, the eliminative materialism project could turn over a rock, finding out some key fact about how our brain works, and wind up showing us that the entire free will project is simply a big misunderstanding.

Rather, what I'm trying to say is that it's through our intuitions that we understand what free will is and isn't. Much of the main debate on free will is about characterizing what those intuitions are telling us. And maybe libertarians are confused about those intuitions; maybe compatiblists are; and/or maybe we come with varying levels of intuitions.

The suggestion that "free will intuitions" are by nature libertarian is an old one, but its time has passed. I think properly understood, the philosophical debate over free will has mainly been that of one camp trying to appeal to the other camp's intuition. It's within this environment that PAP had arisen, and compatibilists struggled with mere essays and arguments against PAP, until Frankfurt cases came along.

Experimental philosophy is a bit more objective of an indicator for where the needle is regarding free will intuitions. It should be readily apparent by now that compatibilists too have strong intuitions. And since we understand what free will is by reference, it's important.

Furthermore, LFW and CFW are often viewed as being mutually exclusive; but evolution and our brains don't have to respect the natural side-taking rules of debate forums. Showing LFW viable also doesn't show we have libertarian free will; it only shows it viable. But even if you showed we have something LFW-like, that doesn't "win by default" to rule out that we have something CFW-like. The only way you're going to rule out CFW is to do address the "thing that CFW proponents refer to"