I thought I’d come and check back in here as I’ve made some progress towards finding out the issue, at least. It seems this is a kernel issue, as spotted in this thread about Fedora.
I’ve just switched my Arch install to using the LTS kernel (5.15.43) rather than the latest one and s2idle sleep puts the hardware into the low power mode I’m expecting from it. It seems that 5.16 seems to have introduced this but I’m yet to figure out how to solve that. But progress at least!