Thermal paste does not and will not ever cause the console to just randomly turn off WITH THE EXCEPTION of an overheating event. Yours wasn't overheating unless it beeped loudly and turned the red light on before it turned off.
It sounds like the most likely cause of your original problem is just the typical problem with the solder joints/oxidized pads on the APU. This is a super common problem with PS4's.
As far as the problem you are having now it's most likely just something that didn't get put back correctly (Since the fan wasn't running loudly before you took it apart). I see this often at my shop. It's most likely either that the two large screws that hold the APU tight are not tight, the APU clamp is upside down or something is misaligned that is not letting the APU sit tightly against the heat sink. If I were you I would take it apart again and check everything closely.
I hope this helps and good luck!