You definitely can easily run anything from outside Steam that you've added to as a "non-Steam game" from game mode like you describe; I use this a few mod mana

You definitely can easily run anything from outside Steam that you've added to as a "non-Steam game" from game mode like you describe; I use this a few mod management apps for various games I play, as well as for launching Battle.net on my Steam deck. Not sure if I'm misunderstanding what you mean by systemd rules, but from what I understand, enabling a systemd service (either at the system or user level) is the most reliable way to make sure that something starts up in game mode as well as desktop mode. Because I find it useful to ssh into my Steam deck for a lot of things rather than trying to do it all via desktop mode (e.g. managing config files for Baldur's Gate 3 mods that aren't from the in-game mod installer), and I always turn off wifi on it whenever I leave my apartment with it to save battery, I enabled the sshd systemd service, and it's available even if I boot up directly into game mode. I haven't actually been able to find any other way outside of an enabled systemd service to make things start up automatically in game mode, which ended up being an issue for me when I wanted to autostart something through Wine that didn't have an option to avoid trying (and failing) to open a window.