The fact that pushing the bars forward, or pulling them towards you seems to take the wobble away makes me think it would almost have to be in the steering neck assembly. I would put the bikes front tire up against a wall, and push the bike forward, and see if you can see any thing moving, or wiggling, or maybe just get on the bike, or have someone else get on it, and hold the front brake hard, and try going back, and forth while looking for slop. I don't see how pushing forward, or pulling back on the bars could have any affect on a bent, or badly imbalanced wheel, or bad wheel bearings, warped rotor, or really anything bellow the steering bearings.
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