I'm on mobile so can't easily pull up an example part number, but digital signage controllers can often be PoE powered. They're insanely overpriced new from the actual suppliers, but for hobby projects they can normally be sourced relatively easily on ebay. The trick is that many of the ebay sellers don't bother listing the specs, so you need to first search digital sign cintroller/computer on ebay then look up the spec sheet from the model number.

Another annoyance is that you need to buy a hat for PoE, it seems like an oversight

True for base PoE (802.3af, 15.4W), but if you have PoE+ or greater (802.3at, 30W and up) you can start to power more common PCs - I’m running a couple repurposed Chromeboxes from PoE++ adapters.