Edit: I’m glad so many of you have had no issues with multiple monitors. My set up is a little unusual (3rd display is an infrequently used large tv hooked through the receiver) and is definitely solveable but will take some effort (and honestly, I’d rather spend my spare time outside or with friends, so who knows when I’ll fix it.)
I have a similar use case with my PC and TV. My PC is across the house from the TV and is connected via an HDMI over Ethernet KVM for when I want to use my PC as a gaming console.
What I ended up doing was creating an automation in Home Assistant to turn on my KVM via a smart plug, then wake-on-lan my PC, and intiate a Steam Big Picture mode gamescope session. This was pretty tedious to get working all together, and startup time is pretty abysmal (around 1 minute to get fully into Steam), but it does actually work consistently.
In case anyone is interested in replicating my setup: I’m running NixOS 25.11 with the Jovian flake installed, and launching my session via the systemd service
run_gamescope. If you’re not on NixOS, you should still be able to build your own solution by emulating the Steam Deck startup services (honestly, it’s not that complicated), or looking into projects like ChimeraOS.