My HP All-In-One 20-c081nt has the processor Intel Core i3-6100U, which is supposed to not run hotter than 100C. On Windows if 100C is reached, the screen will fade out and PC will immediately shutdown. A warning will be shown at next boot. On Linux, seen in the video, the PC will simply keep running as if nothing has happened and show the thermal shutdown warning after a graceful reboot.
Unless the heatsink is entirely disconnected it shouldn’t power off instantly. Something is very wrong with your machine and I don’t think it’s the over heating protection. I’d remove the heatsink and replace the thermal paste and make sure it’s seated fully when you reinstall it.