Oh yeah, while you have it up off the ground, check the wheel for play. Have a friend hold the bars steady and you twist and pull on the front wheel checking for play.
If you take off the front wheel, check the wheel bearings for rotational smoothness and play. Also check the axle for straigtness. Simply roll it on a smooth flat surface, if it is bent it won't roll.Do the brakes drag? Does the wobble get worse under braking? Check the rotor bolts for loosenes or missing bolts. Check that the rotor mount has all its floating pins and the rotor has no excessive play.
That is about all there is on the front end to check. Bottom line is that the damper is there to catch any rider induced forces that can destabilize the steering. It isn't there to hide a problem with the front end geometery and servicing that causes a wobble. The bike should ride straight on a flat road without hands at any speed with no wobbles. Given no head bearing problems or damping/preload misadjustments, it is the tires causing problems. Try the Metzlers M1 or Dunlop 208's
Has this bike ever been in a front ender? If so maybe the steeringhead is bent sideways? if the front and rear tire are not tracking identicle lines, the bikes tires will be fighting each other to keep itself balanced and might shake at speed. You would see evidence of this in tire wear. each tire would have a feathered wear pattern or cupping on them, but the pattern would be opposite direction on each tire. This may cause the bike to feel like it is riding slightly sideways down the road too. Sideways bent forks, unequal spring preload, unequal damping, or bent axles can cause this to happen also.