... But then don't you have much less grip, as you reduce the width of the tyre?
Agreed, with all other things being equal, you might expect 10% less grip.
But although the width has decreased, the contact pressure has increased by the same amount, and you are down to the physics of the rubber about how this affects the grip.
And there are times when more rubber is recognised to be detrimental (snow, standing water, etc) so "more rubber = more grip" does not always hold true.
And this depends on all other things being equal; the post-facelift R170 has an additional tie-bar (anti-sway bar) on the rear axle; for a pre-facelift vehicle, this factor may exceed any benefit that a different tyre design could provide to tighten the response of the car going though a series of corners.
And seriously, how often do you drive your car that close to the upper limits of what it is capable of? I suspect improved feel (by reduced sway) may come into effect much sooner than ultimate grip. Of course, hazard assessment would say the danger of failure of ultimate grip is greater, so the ultimate grip is more important. (Poor sway characteristics probably cause the driver to go slower, too, so improves the safety by *reducing* confidence!
There is a much quoted comment from Colin Chapman that tyres produce maximum braking force at 10% slip; getting one tyre to that optimal configuration is such a challenge, let alone four, that it takes a computer to control each of the wheels independently, and was unrealistic in the (pre-computer) real world. But I do wonder how much this braking force value varies with condition of the rubber, brand of tyre, sizes, configuration, etc, in terms of the maximum braking that can be achieved for a car.
As for tyre size, that's right, 225/40R17 for the back for R170 AMG... I am changing them tomorrow actually, and was wondering the best compromise grip/road noise... I had some P.Zero in my previous car, but they generated so much road noise, and I changed to M.Pilot Sport, it was a bit better but still... I wonder if the Yokos would be OK...
I have a personal bias towards Pirelli, since I have found they work with all the cars I have pushed hard, but I know that for similar cars, this doesn't work. (Vauxhall Nova v Ford XR2; similar cars, but on P6, the Nova was a joy to push in and out of drift around corners, and the XR2 had no feedback until it let go)
And I had Michelin Pilots on my Clio, as specified by Renault, both HX and SX, and neither were good enough for the brakes, and I finally got the braking to work properly when I moved to Pirelli P5.
But tyre choices seem to be a combination of personal preference and how the car is set up.