Well, right now the weather forecast is cool, but sunny.
Core Temp is showing temperatures in the high 40s to mid-50s at idle. I cannot remember them ever being this low. Even when things were "OK" the minimum I'd ever seen at idle was in the 70s. This morning when I turned it on to heat up the processor (so it's be easier to get out), it was showing three cores at 99 or 100 and the CPU fan was screaming like a jet. Now I can't even hear it, but am certain it's working. It was momentarily a bit higher when my antivirus was running a startup scan, but Speccy is showing an average during this session of 64 deg.
As soon as I loosened the retainer screws on the heatsink it was clear that there'd be no problem getting it off. I gently twisted (to be sure not to pull the processor out), but there was no resistance. On both the processor cover and the heatsink the old grease was just a gray cake-like film.
I thoroughly blew everything out of the fins on the cooler and cleaned the cover and the heatsink surface with 99% Isopropyl using a a clean microfiber cloth. I applied a "small kernel of corn"-sized dab of Arctic Silver 5, aligned the screws, tightened (and said a prayer).
It booted faster, with no immediate screaming fan: that's when I knew it'd be OK.
Thanks so much to everyone who weighed in: I could not have resolved this without your knowledge and help.