I have used Copyparty on my LAN (with nginx as a reverse proxy), and remember it being fast enough to saturate a gigabit connection. I would probably believe the numbers quoted here: https://github.com/9001/copyparty?tab=readme-ov-file#reverse-proxy-performance

Does the server have to run headless, or can one person still run games locally on it while another has a remote session? That is, would I be able to play on the monitor/mouse/keyboard directly attached, while my partner has a session on a laptop?