"Well, they pay the songwriters. Aren't they artists?"
Yes. And so are the recording artists. And so are comedians. Why shouldn't they be paid when it's their performance of that written song or comedy act that is making the radio broadcasters money? Nobody would make this argument for any other job. Why are people making it for musicians?
"And is it really so outlandish that the writer gets paid from radio appearances and the performer gets paid by, you know, performances?"
Yes, it is outlandish that recording artists do not get paid for the publication and distribution of their recordings, which make the radio stations money. It is very outlandish. People should be paid for the work they do. This is not the controversial position you are making it out to be.
"Musicians who make good music don't get into it for the money, because it's so unlikely to ever make a lot of money even if you're very talented."
You could say that about anything requiring talent. I guess we shouldn't pay anybody for anything then, because they'll just do it regardless. Because money doesn't have any an impact on compelling people to quit their dreams and go be another cog, somehow.
Happily supporting conditions which make producing art impossible, has a terrible impact on art in a country. But like you said, you don't care about art or culture, so it's "no big deal" if none of them are paid.