That click you mentioned,is it at dead center/straight ahead in the handle bars? If so,the bearing race has wore a spot in the center causing that click. They're kinda famous for doing that actually. IF it does have that worn spot in the middle then what can happen is you have to overcome that click to turn and it causes you to sorta flick the bars and causes it to feel like its wobbling, The faster you go,the worse it feels. You get the same sensation if you overtighten the steering stem. Ask me how I know
The effect is subtle that you cannot even tell you are doing it,but it does happen just the same.It just makes it impossible to make the slight steering input needed at speed to maintain a straight line-it forces you to over-steer,whether you notice yourself doing it or not. If you click is what I think you described( or what I described?) then that sound like your problem.