as far as I know, there is still such a thing as fair use--you are entitled to record broadcasts and music for your own use
Yep. "Time-shifting" is established as fair use.
and to share those recordings with friends
Nope, that's never been fair use. Distribution is distribution, whether you're distributing it to friends, enemies or people you wouldn't know from Adam. You're allowed to distribute parts of it as quotes - for educational purposes, for example - but this has to be just a quote, and the fair-use limit is usually 10% of the whole word. (And no, you wouldn't get away with splitting it into 10 separate files and saying that's fair use! ;-)
I have never heard a writer demand money for that kind of performance.
Except they do, and that's what the PRS and similar groups are set up for - so that you get paid by people doing covers of your work.
Which is why I think they've got their heads up their collective arses in trying to squash online guitar tabs. The harder you make it to learn how to play your songs, the less likely it'll be that anyone will learn your songs. And that cuts how much you earn from your writing.
Scrump, you're right that you could do that. However, you then need some mapping between your new song and the original, otherwise it's useless. And the moment you do that you're screwed.