The processor board definitely can shutdown the Mac.
I had a faulty CPU tray about 2 years ago. I use my Mac Pro since early 2009. Basically leave it on 24/7, never do any maintenance for more than 5 years. So, I guess the dust cause extra heat accumulation and eventually kill the north bridge. It's very dusty in my city, and I often leave the computer 100% load in a room that with no air con. The room temperature can easily go over 30C. Northbridge often stay at or above 80C.
The problem start with few self reboot, self shutdown may be once or twice (when I leave it alone running handbrake), and occasionally freeze. These symptoms occur less then once a week. So, a bit annoying, but not too damaging to me (my computer is for leisure use anyway. Initially, I thought may be my room temperature is too hot, and the W3690 run hotter than the stock W3520. So, the CPU was overheated. Therefore, I create a new fan profile to keep the CPU at or below 78C. It alleviate the problem a bit, no more self shutdown / reboot, but still occasionally freeze.
Until one day, it self shutdown when I was not there. And when I try to switch it on, power up, fan spinning, but not POST, nothing display. Took the cMP to Apple. They try all PSU, GPU, Logic board, nothing help. And I request them to get the CPU tray for testing. They did it, and a new CPU tray fix the cMP in no time.
I know every case is difference, my CPU tray is faulty doesn't mean that yours also. However, from my experience, the Apple staff just follow their manual. Which suggest that the 3 main cause is PSU, GPU, and logic board. CPU tray is not on the list. So, if nothing work, you may ask them to replace the CPU tray, otherwise they may not consider it.
That's wasn't too bad for me. ~$300 for everything (cleaning, testing, parts, labour, 3 months warranty). Anyway, I learnt, even the best designed computer need maintenance. So, I shutdown the Mac and blow the dust away about 2 times a month. It's 2 years now. Not a single self shutdown / reboot / unattended freeze (software crash of cost can happen, but that only happen when I was using it and trigger the bug in real time.)