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.

An old micro pc from dell/hp/lenovo. They are often cheaper and more capable than Raspberry Pis. You can just put up a random Linux distro and it will work.

If they have an Apple TV, you can just install the app and use it as an exit node. I would check out the devices that are on their network currently, chances are you can use one of those.

Use a pi zero it’s like $5