I can't comment on MB's on cheap tyres since I'm relatively new to MB ownership.
However, about 5 years ago I bought a used BMW 540 fitted with Falkens with about 5/6mm of tread all round. The car has 18 inch Alpina wheels, 235/40/18 on the front, 265/35/18 on the back.
After driving for a while I realised that it felt fine in the dry and terrible in the wet. So much so that on a motorway trip in the pouring rain I actually pulled off, turned round and headed home, it scared me so much. I took the car to a local BM specialist to find out what was wrong with it. He gave it a good check over, told me everything was fine and advised fitting some better tyres.
So, I had Bridgestones fitted all round and could not believe the difference. The car now felt totally planted in the wet.
Frankly you couldn't pay me to fit cheap tyres to any of my cars!