Most of these containers don't require any thing attached to the host - it can work headless. They are docker (just normal Linux inside) containers which are running processes such as Firefox and usually run something like KasmVNC giving fast, responsive (if using high bandwidth settings) remote desktop into the containers which runs the Firefox browser on a lightweight virtual desktop with a virtual display. I'd recommend looking into the concept from linuxserver.io's Webtop container to understand the concept (and how cool they are!) In your case with obsidian they pretty much run a container and pass a bind mount for a Vault. This isn't synced by default with anything but you could use Syncthing on the host to sync your vault from other devices or use Obsidian Sync.

I have used Sunshine + Moonlight over fibre + 5G based networks and its worked pretty nicely before. This was over Tailscale and routed over a DERP relay (worse latency, more hops due to CG-NAT). As long as your host internet speeds has good uploads and usable latency as well as your server/desktop connected wired to your router over Ethernet then it usually works well. Does depend on your network but it does work (and pretty well, as long as everything is configured properly), but YMMV.

wondering this also! wouldnt it require a domain for your account though?