Laptop Overheating Oddly

I am working on a friend's older Toshiba laptop, it is Intel I7 based. She bought it 4 or so years ago. It had worked fine until recently. It is apparently shutting down after being on about 5 minutes due to high CPU temp, my best guess from a bunch of troubleshooting. When starting work on it actually seemed to work for me a couple of months ago. She took it home and brought it back then telling me exactly what happens. It just simply shuts down in about 5 minutes. I removed the case and found the heatsink/fan with a lot of dust on the fan and then complete blockage of the heatsink fins which I cleaned. I figured it would work but it failed again, maybe staying on a bit longer. I could feel the heatsink on the CPU grow hot yet the fins never got very warm. So I bought a used replacement heatsink and installed it. I used a heatsink compound at the CPU. It then seemed to stay on 3 or 4 minutes longer but ultimately shut down again and the heatsink at the CPU was very hot. The heatsink tube does seem to be warmer further along than the old one and the fan is pushing lots of air through it. I will have arctic silver paste tomorrow to give it one more try but other than that I really can't figure it out, it runs fine otherwise when it boots into Windows. These tests were done simply with the BIOS screen display under idle with no hard drive. Could the CPU have been damaged so the thermistor circuit is not sensing correctly. If the arctic silver does not solve it then she is going to have to get a new laptop unfortunately.