In steam deck settings, there's an option for external monitor safe made. This limits resolution/refresh rate/hdr to improve compatibility. You can also try manually limiting external monitor resolution.
I know I had an issue where an slightly older TV I had started rejecting signal inputs if the resolution was too high or hdr was on. Originally I thought my deck's video output was broken, but I tried a 4k chromecast and the TV couldn't display it either (despite being a TV that had previously supported 4k and HDR). Limiting to 1080p with HDR off made everything work again.
Yeah, I've gotten sucked into HSR as well (first f2p gacha game that's ever hooked me), and I can't imagine having time for another game like it. The daily content isn't really any trouble to do, but having to do that for multiple games would get old fast. And the monthly content drops can be really substantial and take a long time to experience all the content.