Shit man, I dealt with anxiety and depression for maybe ten years doing mostly hospice/end of life care. Acting is a difficult profession (seriously, it is) in a lot of ways, but it's mostly controlled. They have schedules, and the actors rarely have to deal with the crises of making the shoot happen (except the ones silly enough to try and direct or produce lol).
Any job can be done with moderately controlled anxiety. The only question is how long, at what price?
Since those folks get paid way better than I did wiping asses and moving bodies, I suspect they can call their therapist or whoever during a major anxiety event, and can afford the usual meds to help ease them along.
Musicians are similar, once they're big enough to do big tours. They have managers and assistants to help with the most stressful things outside of the performance itself. So they only have to worry about part of it.
Besides, I dunno if you've ever tried and performance of any sort, but it's a different kind of fear. Yeah, stage fright exists, and everyone stresses about doing well, but people that pursue it as a job get a bigger internal reward out of it than a filthy casual would. So the payoff of the stress is higher. Like, for me, I'd never be able to do theater with any regularity because I don't get that joy from it. But someone that's top tier and doing it for a living definitely get a big dump of dopamine and other neurotransmitters from it. People think I'm fucking crazy for some of the martial arts shit I used to do, but it fulfilled me, so the price was worth it. Same thing for them