[UPDATE] [SOLVED]
So, as I have some IT backup and no time for any other troubleshooting (and even less for tinkering) as I am on working schedule, I just left the machine unplugged.
I plugged it 5 hours later (at my lunch break) and 20 minutes later it was stable (-ish) around the PID set time. I pulled 2 coffees and everything remained normal.
So, looks like some electronic shenanigan going on there, but old school IT troubleshooting fixed.