I tried the idea of varying the yTran with the xRotate, but it just does not "cut the mustard". The problem is that for a realistic motion, a point at bottom center of the chair should move slightly backwards and forwards relative to the floor as the chair rocks, but using yTran this point just moves up and down in a linear way, and this also puts the rails at the wrong altitude at some rotation values.

To put it another way, you are essentially rolling a wheel when you rock the chair, and when you roll a wheel it moves backwards or forwards.