Well, the dryer has been disassembled and vacuumed, the dead coil removed, the new coil swapped in, ready to reassemble as soon as the new belt arrives. I was surprised how much of the brown shmutz in the heater was just ash from the coils - it dusted off quite shiny after I replaced the coil.
Before
After
I'm a little worried about the visibly duller part of the heater assembly, hopefully that won't absorb too much heat - I tried to cluster the coils a bit looser there to compensate.
It's even visible on the reverse side.
Either way, I'm reassembling tomorrow when the new belt arrives.
That's what I'd assumed. Wipe it out with something lint free to make sure there's no grit, then add the new grease.
Some greases are incompatible, and not cleaning thoroughly causes them to react chemically and cause issue, but unless you're in that situation, you don't need to clean super well.