yeah, a cursory google search of the symptoms points to... ram speed issues?? but i don't know about that. seems pretty far fetched. it still seems like a state issue since the power button is unresponsive. i would almost measure the current of the computer being "off" (ie waiting for power button to power on) and "shutdown" (by linux) and see if there is a power consumption difference to see if it's just sleeping of if it's actually shutdown.
maybe have a play in the bios in your power settings and try disabling or changing any kind of power option you see one at a time and see if it start operating properly changing one of those... S3 sleep state to S4 or S5 if available etc. or just disable everything.
https://learn.microsoft.com/en-us/windows-hardware/drivers/kernel/system-sleeping-statesi know this is for windows... but it explains the different sleep states a computer motherboard can be put into. this issue is weird enough that i still think the computer is being put into a weird sleep state where it's waiting for...something... but seems to be ignoring the power switch. which is why i nitially thought it might be waiting for a "wake button" press
maybe see if issuing in a terminal window a "shudown -h now" command and see how it acts... does it power off? does it stay running? ... in any case then reboot the machine and then issue a "shutdown -P now" and see how it acts trying to power on.
the -h parameter instructs the system to shut down and then halt.
the -P parameter Instructs the system to shut down and then power down.
just kinda throwing things out there and see if we can see a weird action from the system to narrow down what the issue is.