My understanding is that Flatpaks are generally reserved for GUI applications and not command-line tools or servers. I’m not sure it’s the ideal format for Immich.
That said—and I’m probably going to get hate for this—there is an Immich snap package that does just what you’re looking for.
Star Labs is a European company that sells laptops and two-in-one tablets with excellent Linux support. They don’t meet all the criteria you listed, but they’re good to mention