Your server guide indicates that you can't run the dedicated server on linux or headless. I currently have a server running on a headless linux miniPC through wine.
It's using bog-standard system wine on Mint 22.3 and the only thing I had to do was add vcrun2022 to the prefix with winetricks. You also have to tell steamcmd to force platform type to windows for the download. I then use a systemd service to keep the server running and use cron entries to update and reboot automatically nightly.
I understand that y'all might not want to support this, but it does work. Maybe update the guide to say that it can work but you don't officially support it? I've attached my scripts and service file so y'all can see exactly how it's done.