I did not remember who a W124 would drive so nicely and true until I had a major faclift to my suspension/steering system.
At 235K, I replaced the 4 shock absorber and almost all rubber bushings in the suspension/steering system, along with 2 ball joints.
The road behaviour of the car deteriorates the first day it goes out of the factory. But it does it so slowly no one can see/feel the deterioration. After the job was done, I could not believe or remember my Benz being so stable in all conditions. A real bullet train. No outside force can move this car from the intended track.
Your problem (like mine) is most probably due to a combination of factors (age and mileage related). Tire, steering components, suspension, etc. Identifying THE cause might be difficult