Can I check if the cooling system of my Macbook Pro 15" Early 2013 works as intended? I have this model: 

MacBook Pro "Core i7" 2.7 15" Early 2013
2.7 GHz Core i7 (I7-3740QM)

Intro.  February 13, 2013   Disc.   October 22, 2013
Order   ME665LL/A   Model   A1398 (EMC 2673)

Recently it started shutting down randomly and I took it to a local repair shop. Tech guys told me the problem was with the power controller board (because of the constant overheating) and that they had fixed it. However, the cooling system (the radiator itself) was failing (probably out of gas inside due to them) and needed to be replaced. They failed to find a replacement part (since the laptop is old) and recommended me to restrain from overloading the system in the meantime. 
The laptop is now running smoothly. I'm measuring the CPU temperature with Macs Fan Control and it is showing 103 Celcius degrees on the CPU cores under the heaviest load possible (which according to the tech guys is a way too high). However,  according to this answer:

https://apple.stackexchange.com/a/284437/330383

there is an Intel defined spec for my CPU:

https://ark.intel.com/content/www/us/en/ark/products/70847/intel-core-i7-3740qm-processor-6m-cache-up-to-3-70-ghz.html

The Tjunction value (value when the processor starts to slow down to avoid damage) is 105 Degrees. 

I never measured the CPU temp before the incident but I feel it was always running this hot from the very beginning.

So, please help me understand - did the radiator really fail or it was just a random power controller problem? 

One more thing: I was using a faulty power adaptor for some time which had an overheating connector - THAT could have possibly caused the damages to the power board. To my mind, it may never have been connected to the CPU overheating.
a faulty charging cable can cause all kinds of problems. That could have started your issue and it may have grown from there as it is what is sending power to all of your macbook, it is not limited to what it can break. 

one question I have is what are the circumstances for the use of the laptop. meaning are you using the laptop on a desk? on your lap? 

you can always get a laptop stand that has fans built in, and will help cool the system, and keep it from overheating due to being on your lap, or a blanket, something like that. 

the temps you are seeing are pretty high, here's an article with some people discussing the temps in their MacBook of that year.

as far as finding parts, I would suggest talking with the best MacBook experts on the planet at Rossman Group as I'm sure they can find the part, and will be sure to diagnose any problems and where they stem from. I spend quite a bit of time on his youtube channel watching him repair boards, and other issues, he is one of the true original MacBook and apple product repair geniuses. Plus he's pretty funny with his rants.