If you use Poe you need 1 cat6/device. Hacking cat6 to carry dc current like you said is likely out of the question if the devices aren’t nearby. You will pick up and emit a ton of noise from that cable otherwise. At the end, putting cat6 everywhere you need, and all the other places you dont need it is a good idea, whether you have lots of poe or not.
I was in your shoes 8years ago rebuilding my home (before me being into home auto). My buddy electrician recommended 2 cat6 to every light switch… I didn’t end up doing that because ‘overkill & cost’… just did 2 cat6 2 rg6u 1spk pair and one telephone to most rooms (office/workshop areas got more). I really regret not listening to by buddy honestly (did all the work myself so wouldn’t have REALLY cost me). Cat6 oozing everywhere not just for Poe devices, but also control of alarm and whole home a/v you get lots of options. You said it, wireless sucks… how much is a few more boxes of cat6… a few $100… you get options you’d have to pay $1000’s for to match with wireless.
Also, definitely dont forget to run cat6 to any place you think a camera may go… and places where a camera shouldn’t go lol. This will be a huge bonus if/when you add cameras. Bottom line, poe cameras tend to be good (like protecting banks good), other cameras tend to be more like toys.