running Intel Atom processors[...]these were full-fledged x86 computers that required more power than what the standard PoE (802.3af) could deliver
Those must've been the server Atoms or the later models that aren't actually all that low-power, as the ones I'm familiar with are well under 10W.
You might have a 10W TDP CPU, but the rest of the system requires power too.
What I don't understand, though, is that these were for "digital signage systems" according to TFA. You're not running Windows 10 Pro on an Atom board and a large illuminated digital sign with 25W PoE. Maybe the signs were smaller than I'm thinking? Like tablet-sized? But if you're needing to run power for a display, why not just power the whole system from that?