Turns on and works fine for 3 minutes, but shuts off unexpectedly

Ah! You'll need to inspect the thermal paste quantity! I often see non-running systems often is caused by overloading the TIM getting it under the insulator getting the small SMD components wet with the TIM which messes up the timing logic. After removing the logic board lift off the heat sink.

This is too much!

Give this a try get a single sheet of face tissue cut it into the same size as the CPU and then wet it making a tight ball. Even this is to much! But it gives you an idea how little you really need.

Now the hard part! Getting the TIM off! After carefully pulling away as much as you can physically, Then use a small soft toothbrush (kids) to help. After I've pulled the old paste away I use a loadable spray can which I load up with solvent (Give 95% Isopropyl a try first, you may need something stronger). This is what I use True Power Rechargeable Sprayer It allows me to get into the nicks and crannies between the SMD devices and under them a bit.