It's easy to abuse the position of spending other people's money to support something as nebulous and subjective as art
No it isn't. It's extremely hard, and is fought for every time these things come up for review.
Personally I'd consider the world to be better off without much of modern art, by what right are my taxes spent on such things?
They used to say the same thing about Matisse. Or they would have, if grants had been around then. In any case, the amount of money spent on the arts is microscopic when compared against any other public spending. It's pretty cheap, and it's good for everybody.
Today though we've created the strange idea that an musician should be able to record their music once, and get paid for it repeatedly over the course of the next century
I have often thought the same thing. But if the company I worked for was not able to charge, over and over again, for the software that I write once, then the company would not exist, and I would be out of a job. There's nothing wrong with expecting payment for a copy of your performance. Not for a century thereafter, sure, but for a reasonable period of time. Twenty years seems reasonable.