In the Steam Link app, you have the option to select "Start Streaming" without picking a specific game. This will stream the screen as it is without binding to a specific window.

The main caveats here are as follows:

• Requires a working pipewire & desktop portal configuration
• Depending on desktop portal & settings, you may need to manually click through the screensharing request modal on your desktop at the start of each connection
• The Steam client must be installed and running on your Linux machine in order to receive connections