In quite a lot of cases they do (or at minimum the right to reproduce and distribute such at will without compensating the originator of the work).
https://www.trademarkandcopyrightlawblog.com/2012/05/bear-an...
https://designobserver.com/feature/who-owns-student-work/126...
https://www.huffpost.com/entry/who-owns-student-artwork_b_64...
My mother got hit with this when she tried to patent the seamless sweater invention she had - suddenly the university stepped in and said it was their invention. Sure enough, she'd signed over all copyright when she started the course.
but just because it happens now doesn't mean it's morally correct. In an ideal world, do we want this?
"Do I try to make my career defining art now or do I use this time to explore a variety of subjects and mediums?"
The fact the artist doesn't own the copyright to this early work tilts the question to the latter statement.
To answer your thought experiment: Should an Art Teacher get copyright rights on all their students' work? (what you propose is effectively art college)
Agiain, to my mind the problem comes when someone (let's call him The Baron) employs a hundred Devs to make an AI that provides cheap legal advice, based on public-domain court records. People get cheap legal advice (society wins). The Devs get paid for their professional labour. A million lawyers go out of business. The Baron becomes astonishingly rich.
Should The Baron get all that value? Should the Devs? Should the out-of-work lawyers?