Tangentially, I was also curious whether a thermal paste can also be electroconductive if used not as intended. Can a thermal paste, especially on copper or metal oxide basis, also be electroconductive?
[1] https://www.ifixit.com/Answers/View/723317/Optimal+repair+of+HP+250+G6:+overheat,+cooler,+display+frame#answer723320
Maybe clean the fan, but I mostly suggest to always replace it if the form factor allows for it if you open the device anyway. It may have a broken bearing, which in most cases isn't that bad aside from the additional noise. So it is strange that the notebook seems to overheat. Perhaps there is really some form of blockage which just needs to be cleared.
Does it overheat running on battery and/or wall power?
Do you have a plan if you loose/strip screws and cannot reassemble the computer?
Good luck.