The reason why we would get a world without professional musicians would be that there was a vast surplus of musicians leading to supply massively outstripping demand. Which means we'd have a world with a lot of amateur musicians playing for fun, without pay, because they enjoy doing it. Much like we have today.
To answer the question of what the GP does for a living, one can assume he's doing something that nobody else would do for fun or as a hobby. Probably not trying to be a professional starcraft player or a pro angler. Much like most people who hold paying jobs, making the choice between fun or a paycheck. It's not a question of the value of the work, it's a question of wether the value is available anywhere and everywhere without remuneration or not.
The reaction from 'professional musicians' when Amanda Palmer invited amateurs to play with her on tour locations is an exceptional example of what must be the most spoiled and conceited group of people ever to grace the labour market.
You're correct to say that in a world without professional musicians, there would still be a lot of music to listen to-- and some of it would indeed be great. But let's consider for a moment what else this would mean.
Do you like listening to classical music? Kiss it goodbye. It's simply not possible to play classical music at a world-class level without dedicating yourself to it full time. Itzhak Perlman once remarked that when he misses a day of practice, he can tell that he's not playing as well; when he misses two days, his fellow musicians can tell; and when he misses three days, everyone in the audience can tell.
Got any records at home? How many of them do you think would have been brought into existence if the musicians on them were doing it merely "for fun, without pay, because they enjoyed doing it?"
You mention the Amanda Palmer incident; you may want to read her own comments on the matter, which are more thought-out than your own. Palmer didn't specifically solicit "amateur musicians" for her shows. Instead, she posted ads for musicians (who were expected to perform at a professional level) and offered them "beer and hugs" instead of pay. Her rationale was that-- well, sometimes professional musicians are willing to play for little or no money in return for commercial exposure, she'd done it herself early in her career, and occasionally still did. But she also admitted that she was wrong. She apologized for not thinking things through and promised to compensate her sidemen in the future.
(Part of what irritated everyone about the Palmer incident, of course, was that she has become a well-known and well-compensated artist, who had in fact boasted on her blog about "raising $1,000,000 on Kickstarter faster than any musician in history", and so forth. She's not a struggling artist any more.)
The labor market for musicians has always been tight-- so tight, in fact, that 99% of aspiring musicians will not succeed in paying their rent. The issue here is that in a tight labor market, it's especially important to make sure that the boss is treating you fairly and that there are no middlemen in suits ripping off the modest wages that you are able to command. Read up a bit on the history of record labels.