I have no need to do it on my Mercedes but I have owned 2 vehicles that if you refilled the coolant would trap Air behind the Thermostat.
I drilled 1 no bigger than 1/8" just so that the Air would pass I could fill with Coolant without the burping issue. I did not do it to solve any overheating problem.

I also used to Test Thermostats before I put in a new one.
After changing Head Gaskets on my Van I could not find the original good Thermostat and bought another one. I did not test it. And, sure enough I paid for that.
I stopped the Engine just as the temp was approching the Red Zone.

Had to remove the Thermstat and test it and and sure enough opening 10-12 degrees late (the stock Thermometer tem is 190 degrees F so opening 10+ degrees late makes a puts it on the edge of overheating).

Suspend the Thermostat off the bottom of a pot of water (with a tsp of Salt disolve in it so it does not bubble so much). Put a Thermometer in the water (Candy or Meat thermometers work) as the temp builds up watch when the Thermostat begins to open and note if it opens fully or not.
I am OK if it is within 5 degrees of what it is supposed to be. 10 or more degrees is too much.
Because I did not test the new Thermostat I cost Myself a lot of extra labor and mess of having to Drain the Coolant and refill it again.

Another Note: When I returned it the Counter Guy tried to sell me a lower temp than Stock Thermostat. He could not get the idea that if I was overheating with what is supposed to be the correct temp Thermostat that I should fix what was causing the overheating.

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84 300D, 82 Volvo 244Gl Diesel