It's kinda... expensive to add. Proper POE is 48V so need some bigger pin spacing and you kinda want to have galvanic isolation too and that's some extra parts that are not small.

Just look how big even the smallest rPi POE shields are (and I think some of them don't do the power isolation.

Also, realistically if you're also not booting from network you want some kind of UPS to not get dirty shutdown every time. I've used UPS Pico which has option for supercap but feature-wise it is a bit overkill and also doesn't do isolation nor power negotiation, just plain 24V passive PoE.