Something like LPS/TENS would fit your needs exactly, but it stopped being supported in 2021. It was essentially Tails but was built to be run from a wide verity of sources (we used to network boot onto PCs that had no hard drives at all, but had local network share access).
You don’t need Linux for Plex or Jellyfin, both have prebuilt server installers for windows. If you want a general guide, Jeff Geerling has a video from a couple years ago about setting up Jellyfin on I think his NAS. But honestly I would just go read the docs for the project you choose and follow their install guide. You’ll need to ensure your media is organized on your drive, but you should do that anyways. Other than that you’ll need a PC that’s essentially always on (or at least is on when you want to watch videos), and a client at your TV like an Apple TV, Android stick, or FireTV. I have run both, and Plex is probably the easier, just works, solution, but it is closed source and needs a subscription for some features (though you can use it just fine without one). There are some hiccups I ran into primarily around streaming to an Apple TV, but honestly it was super stable for me. I switched to Jellyfin recently and the setup was a tad harder and the UI is IMO significantly worse than plex, but it’s open source and fully free. I may switch back to Plex here soon just because their AppleTV Live TV (from my antenna) is just much better supported than Jellyfin.
The services often associated with these media servers that are designed for searching and grabbing content are the *arr stack (radarr, sonarr, etc). You will often see them talked about in the save places but you don’t need them to run Jellyfin or plex.