A couple of things to try or to think about.
After you have ridden a while, you may find that traffic or some idiot in a cage had crowed you into a curb so that you are running about 30 MPH with your tires rubbing the curb. This is a very difficult thing to get out of without a crash as you cannot counter steer as the curb is there and you might be already actually having your weight too far over. You will need to lean aggressively and quickly because you can’t turn the front tire away from the curb as that will lean the bike even more ‘over’ in the curb direction. You have to use weight shift alone if you can’t get stopped before you ‘high side’. The front wheel ( handle bars ) must be left alone and you need to hang off the ‘road’ side fast and if that idiot in a cage still there, it can get even more dicey. Once against the curb, you cannot use the bike to get the center of mass back where it belongs.
Second. With practice you can dodge a skunk in the road without you actually moving your upper body much in a lateral direction.
If you do this where you can go back and look at the tracks you make, you will get a real good understanding of why M/C’s steer like they do.
Tracking straight at a 6" manhole for instance, you can flick the bike to, say the right, by starting an aggressive turn to the right. You will ‘push left, or pull right’ , which ever way you like to think of it and the wheels of the M/C will move to the right of a straight track and the bike will start to lean but as you feed in a bit more counter steer, the bike is still traveling in a more or less straight line, actually moving out from under you to the right, let it do so. By this time you are actually past the manhole cover and with counter steer in th eother direction you stand the bike back up. Your wheels will have missed the manhole cover and your bike will have moved right and then left without actually making much or any real turn.
It is just a fast way of going down a street and leaning the bike over one way but shifting your weight the other so that the M/C is leaning at some angle with you holding some counter steer in and using you weight to force the M/C to travel straight by using your weight to keep the center of mass over that part of the tire pactch that allows it to go in a straight line even though you are a bit to the side of the center of the tire.
The real informative part is when you go back and look at the tracks.
Always remember to look where you want the bike to go and not at what you want to miss.
YMMV