I'm not sure what you're counting as "recent".
Something under 3/4 years old.
The PCs I am struggling with are around 7 years old, single core.
OK but you were speaking about replacing them, so I take into account the consumption of the new ones ... Shouldn't I ?
You talk about running machines with a core i3 and SSDs - these are completely beyond our school budget!
Since the "Core generation" Celerons (G530 ...) exist, it's the kind of PC you could have with the budget you were giving (400£), if you chose linux like for the RasPi.
All the way, that's not really the problem here ...
Anyhows, even if I ignore any savings from electricity, I'd still have a desktop replacement scheme costing less than £11K over 3 years instead of £36K. For me the question is to go down this route or something similar, or simply remove the old PCs - I really can't afford to replace them given our budget situation.
OK, BUT the problem is that, depending on your real hardware(*), there are many chances that its theorical cumputing power is already comparable or higher than the one of a Raspi (except for specific things like decoding videos, 3D and so on ...). That's why I do not see RasPi as a real desktop replacement for now in non third world countries, maybe a thin client but it implies other problems/costs.
Further more, for me, an "only" 7 years old PC becomming "unusable", is not normal. There are of course many reasons for that (lack of RAM, virus/bad antivirus, full/fragmented HDD...) but also unexpensive/free solutions. I spend a part of my professional time (for nearly free), making "old PCs" work as they should work, including for schools.
Besides, my main personal (and professional) PC is a bit more than 8 years old today and it's perfectly usable for much of the thing I do (I have a 2 years old linux PC for things that really need a lot of computing power, but I don't use it so much).
(*) : For 7 years old PCs, I estimate something like 2 GHz / 512 Mo / 40 Go / XP