This is basically what I use tailscale & their magicdns feature for. I manage a few locally hosted jellyfin servers for myself and some family members, and its the same problem. I just added tailscale to them all and now I can basically do ssh parents.jellyfin.ts.net or inlaws.jellyfin.ts.net
I was about to say that. This is what I do too.
The only drawback are routes - they won't work on the same CIDR (I mean the fact that you can say in Tailscale "if you want to reach the 192.168.16.13 device that does not support Tailscale, go through this Tailscale gateway"). For this I had to shift my parents' network to be able to access stuff like the printer, in a network that clashed with another one of mine.
The way we did it, roting is not a problem. Any Netrinos client (Windows, Mac, or Linux, including the free version) can act as a gateway. It assigns a unique overlay IP to devices on the local network that can't run software themselves, like cameras, NAS units, or printers, and handles the NAT translation.
Think of it like a router's DMZ feature, but inverted. Instead of exposing one device to the internet, each device gets a private address that's only reachable inside your mesh network.
This overlay approach is fantastic, but I do not think it exists in Tailscale.
I need to implement this type of thing for supporting networks of family members, but without the media server aspect - just computer/networking support. I'm looking for a cheap and reliable device that I can put in each home, to give the Tailscale "foothold". Do you happen to know of any tiny devices? I was thinking there must be something even cheaper than a Raspberry Pi to perform this single function at each location.