I replaced my house's HVAC thermostat cable with one that has more wires in it (5 wire cable replaced with 8 wire), so I could connect the "C" wire and supply power to the thermostat. It only needs 6 wires for that, but since I'm pulling a new cable, might as well have some extra wires for future use.
I tried to use the old cable to pull the new cable, but I messed up and ended up having to pull the new cable without a guide. I wasn't able to do it on my own, but a fox friend came by to help and we got it done pretty quickly.
So now that the thermostat gets AC power and doesn't need batteries, the LCD screen's backlight stays on (dimly) all the time, which I like. And maybe I'll get a thermostat that can integrate with Home Assistant—one that works locally without any cloud requirement. Considering the Venstar T2000, but I haven't researched it too much yet.