It has been running hot for years, and occasionally cutting out. ...
I assumed the cut-outs were caused by overheating, but as the issue appears to be escalating (and can occur when cold)
Warm temperatures seen by computer users do not harm semiconductors. But when semiconductors go bad, heat is the diagnostic tool to make that weakness obvious.
Many confuse this by saying heat degrades semiconductors. Inside your laptop is a failing part. As the part continues to fail, your symptoms occur at lower temperatures.
Warmer room temperatures identified the defect months ago. Now, as that manufacturing defect progresses, the same failure occurs at cooler temperatures. Many techs would have solved that defect by curing the symptoms - more fans.
Better information might be available if your manufacturer was responsible - provided comprehensive hardware diagnostic for free on the disk drive, on a CD, and on the web site. Not all manufacturers are so responsible. But your failure is exactly why the better manufacturer provides those comprehensive diagnostics.
Unfortunately, most techs do not have the necessary technical ability to find that failure- can only swap parts. You probably don't want to spend that much money on shotgunning.