As I understand you get a resonance in the steering (around steering axis) with a frequency of a few wobbles per second. It is usually only a mild wobble but gets worse under braking and the wobble goes almost end to end if you let go hte bars. This can be caused by a worn front tire and the specific thread pattern alone (means it does not occur with every tire). Some bikes (sports with small wide front tires are specifically prone) develop it even with new tires and all patterns. As far as I remember it is referred to in English as shimmy effect and closely related to kickback when riding over a hard, short bump while cornering.
First thing to do is to replace the worn front tire. This will most probably sort the problem.
If you check your steering head bearing the center stad method only reveals a bearing that is too lose or extremely worn / rusted. If the lower bearing is only slightly worn (enough to give trouble) you will have to try and feel the bearing while under load. This is done either with a rotating support pan as used for wheel alignment in cars under the front wheels or a similar device specially made for bikes.
Alternatively you can ride the bike very slowly in first and you will feel the steering "locking" in the center and the line being somewhat difficult to keep.

Do not fiddle with the steering head bearing unless you know what you are doing. The chances that you bugger the bearing are good compared to getting it right.

Checking the wheel bearings is very easy. Ask someone to show you or check in a standard or type specific motorcycle mechanics manual.

The judder under hard braking is a resonance of the front fork. Pretty normal. Frequently seen in heavy bikes with long front suspension travel (like older Transalps and A.Twins)