Absolutely! Any sort of multiplayer game needs a source of authority if you want to prevent cheats like a hacked client lying about its position, and a really good way to do that is load the geometry of your level and run physics checks server side at a lower frequency than once per frame. Godot and Unity both support headless builds for exactly this reason, it's basically the whole game engine, minus the renderer, audio, and UI systems, usually.
That’s complete different. They are rendering the client and streaming it to users. That doesn’t make the client side code “server side” any more than you streaming Fortnite on Twitch does.
