Feels like you're proposing a distinction that isn't universal, and literally there are words (on and off-site) that describe the concept better than the term self-hosted.
Additionally, just because some of the physical infrastructure (the hardware) isn't in your control doesn't make it any less self-hosted. I'm sure there are some that own their building and electrical infrastructure too. Meanwhile I'm a schmuck that has to rent and pay a utility provider for electrical. That's not particularly far from datacenter access.
Plexamp is really great software -- the various methods of randomizing means I'm re-experiencing parts of my music collection like new, and connecting tracks to other tracks that I'd never put in the same playlist.
Go used. Lots of people get rid of their hardware when just a bit of care and repairs will make it as useful as brand new.
Time vs money, right? In six months, I'll forget about the NAS and all the tiny little things I never did / automate. It'll come back to bite me.
With a purchased NAS, that's... less likely? Or at least, I can yell at their support to help me.