Yeah, I'd be inclined since you had wheels and sprocket off to think a spacer is missing or in backwards and allowing travel that's abnormal.
Look down your chain up to the countersprocket - is it straight and the chain alignment correct? Not sure on the Wee but if you get a spacer wrong on the Vee it screws up this alignment something fierce makes it obvious.
On your steering stem bearings, they can still swing side to side sometimes rather smoothly but be out along their vertical axis. To check them for this, with the bike on the center stand pick up the front wheel and pull it out away from the bike. If there's play there, they're fooked. Doubtful this is it tho.
An really out of balance tire is a bumpy wobble that gets worse as you speed...never feels squishy or affects the overall feel of the bike in my experience....more like you're going over a bumpy, rhythmic road.