How can I keep a Foundry VTT server running after reboot without leaving the desktop logged in?

Login and sleep are separate problems. Configure GNOME's power settings so the machine never suspends while plugged in, and make sure the relevant sleep settings are disabled for the server's use case. You do not need to automatically log in just to run a correctly configured system service, and leaving the machine at the login screen is generally safer than starting an unlocked desktop session.

For a home machine that is normally left logged in, automatic login would probably make the setup work, but it is more of a convenience workaround than the ideal solution. If you use it, set the session to lock immediately or shortly after login and still disable automatic suspend. Another option is to run Foundry in a Docker or Podman container, which can be configured to restart automatically and keeps the application environment isolated.

First check whether the service is actually running after reboot by connecting over SSH and using systemctl status and journalctl -u your-service-name. A normal system-level systemd service does not need anyone logged into the graphical desktop. If it is a user-level service, though, it normally stops when that user has no session. You can either move it to /etc/systemd/system as a system service, or enable lingering for the account with loginctl enable-linger username so its user services can run without an interactive login.