Vibration is very different from shaking. Vibration is annoying but does not feel that you are losing control.
Shaking is a suspension problem, not tyres (unless they are very bad).
If it is speed related you can look at the brake reaction bar and all rubber parts. If these are worn the toe-in may vary to much under load. Also check the stabilizer bar, the connection to the top control arm sometimes break.
Tyres are always important for a Mercedes. In my experience Michelin tyres are the best.