Power-over-Ethernet (PoE) uses pins 4, 5, 7 and 8. [...]
Not quite. The PoE standard defines two ways to run PoE over Ethernet cables.
Yes, I am well aware that using 4/5/7/8 is only one way. Actually, I was not aware that there were ONLY two standards here (Last time I dealt with PoE, each manufacturer had its very own ideas about it)
One way is as you say, on pins 4, 5, 7 and 8. The other way is as a DC bias applied to the Tx and Rx pairs. This is one reason you need the isolation transformers on the Ethernet port (somehow I doubt the PHY would like -48V applied to its pins!).
Yeah, I know about the DC bias method. But I won't go that way, as it is to expensive for my use case, and not necessary, either.
Your approach would work when connected to some PoE hubs, but not all hubs.
Who said something about "PoE Hubs"? I'm just going to use a cable adapter to feed the 48V into the unused pairs. I know: crude. But it works. The Power is sufficient good & clean not to disturb the ethernet pairs, and power fluctuations can be dealt with capacitors and friends.
All in all it is about getting power and connectivity to a RPi with as little cabling effort as possible.