Agree with your diagnosis. There are utilities available that can give readouts of various fan speed and temperature readings, depending on the reporting capabilities of the motherboard. It's been a long time since I looked at one so I can't give a recommendation right now.

Many systems have a power supply fan which can easily fail. The CPU fan was mentioned. I've come across a broken clip that let the heat sink come loose. If you do pull the heat sink, you need the special stuff to get a proper mating. A Case fan is less critical but could be an issue. Some higher end video cards have a fan on them.

If it's not a fan or heat sink, then I'd lean to a bad power supply.