What Keith said. "Bump steer" is having the steering angle of the wheel changed by the suspension linkage as it goes up and down in bump or roll, not the concept of having it knocked around by external forces.
I hate it when people abuse established language. I'm still seething over "literally" being redefined as useful for emphasis. Literally literally no longer means literally.
I suppose the actual question the OP asked was how offset would cause wondering (or wandering, and we do seem to have wandered off a bit...), and that's down to, as folks have mentioned, creating varying distance between the steering axis where it meets the ground, and the point from which the tire's contact patch is delivering force. There's lots of fun rabbit hole here, but let's look at the wandering/kickback thing in a pic:
It's fairly easy to see that if this is the left side of a front-driver heading away from us, if we gas it, this tire's going to try to turn to the right, as it tries to use the lever arm of the scrub radius to drive itself around the steering axis (and if we brake it'll try to steer left). Of course, the right side is doing the same thing but in mirror image, and they mostly balance. But the larger that scrub radius gets, the larger the lever arm trying to turn fore/aft (accel/braking/the fore-aft component of bumps and how they tend to add to braking/accel) forces into steering forces. When things happen to only one side of the car, the balance between sides is interrupted, and all of a sudden the steering wheel's jerking in your hands.
Or maybe you drive over a big tar line with just one side and there's just a drag on one front wheel. So you get a pull toward that side. "Wondering."
I'm pretty sure we don't want zero scrub radius, but I don't recall why. I think things get really numb, even though caster/trail would be the obvious self-centering/increasing-with-cornering-load force. I want to say some front-drivers have negative scrub radius, so that when the right slips, the left tries to turn itself leftward in an attempt to balance the offset torque of trying to accelerate the left side of the car.
I have half a mind to apologize for my answer, even though I think it's fairly accurate.