N100-class minipcs are better at everything the pi 5 is doing except perhaps a bit of idle power and gpio.
Raspberry Pi’s biggest strength is its form factor and low power draw.
(plus the screen. And ethernet / PoE variants are rare, and not as cheap, so if that's a hard requirement, maybe not for your specific use case)
Setting aside what they're for, Linux handheld gaming devices are kind of a perfect fit for a minor "house computer". Made cheap by commodification. Flexible. Sadly no GPIO in these I think but tack on an RP2350 and we're golden.
Strip or modify the chassis and embed them hidden or with the screen facing out. Kachinng.
Yes. Generally only requiring a $10 PoE splitter like this: https://www.ebay.com/itm/134500605396
Some N100 class machines draw more power, but many don't, and there are more capable PoE splitters for a few dollars extra.
Use a USB touchscreen.
Edit: Putting a device with permanently attached battery inside of a wall or even on a mount, always plugged in gives me the heebie-jeebies.
Or you look at a mini PC and you really can’t buy one at all for much less than $200 these days. Again, no screen.
But Apple will sell you a refurbished iPad mini for $379 and you’ve got nothing to setup.
I share your concern about running it with the battery all the time, but I think it’s pretty common. I probably wouldn’t put it in my wall but I know of a place of business I frequent often that has one plugged in 24/7 and nothing has happened.
Apple power manages devices that are plugged in all the time, they’ll likely just park the battery at 80%. They are also about as good as you can get as far as hardware quality: Apple sells a bazillion devices and has definitely thought of fire risk.
The other benefit of the iPad is that the accessory ecosystem is vast.