My pro had the exact same "toothpaste" badly applied. I decided to finally clean my PS4 Pro that I purchased in 2017 and the poor console had never been cleaned or opened before up until then.
I took a few good hours to completely clean it out. The 2 games that made my console wail in anguish were Battlefield V and Division 2. They were used as tests basically but I think I prepped well.
My experience is absolutely different from your own as I'll explain but basically, opened it completely, cleaned and reapplied paste using
Arctic Silver MX-4 also purchased a
Samsung EVO 860 SSD (1tb),
Arctic Silver "ArctiClean" (for cleaning the older thermal paste and prepping/reapplying the newer MX-4 paste) and
FujiPoly Ultra Extreme XR-M Thermal Pad (60x50x1.0) for applying to the Semiconductors and the GDDR5. I'm grateful you know just how loud that console can get lol, exactly like an aircraft but I am
thoroughly impressed with the results. It's been a bit over a month and the damned console
doesn't make a sound. It is absolutely quiet and runs extremely well considering I haven't had literally one "CE-34878" error in Division 2 which was absolutely rampant before I opened the console. I believe that may be due to the HDD but I'm not too sure.
Either way, my experience has been that cleaning and then reapplying the higher quality thermal paste and padding paid off in spades and the results are extremely apparent. The jet engine is literally no more

.