Correct me if I misunderstood, but I think you are confused about how the direction of an angular vector can be perpendicular to a wheel's plane if you don't see any movement physically happening in that direction?
Imagine a wheel spinning clockwise in front of you. Relative to your current position, it isn't hard to determine the direction that the wheel is spinning: clockwise! Now imagine that you walk to the other side of the wheel and turn to look at it from that side. From your new point of reference the wheel is spinning counter clockwise. But the wheel didn't change directions. You are just looking at it from a different angle.
So the conventions of clockwise/counter clockwise for determining the direction of the wheel's motion are relative to your frame of reference. The same motion will be described in different ways from different reference points. This is not good enough for physicists, because we need to describe the motion of the wheel objectively, so that calculations can be done and get the same answer from any reference frame.
The right hand rule is the rule that says that the vector's direction is perpendicular the the wheel's plane. This rule is something physicists came up with so they could have a way of describing a spinning wheel in a way that would not change based on their point of reference. It is 100% arbitrary and just a convention invented by us. If there were more left handed people in the world it could have been the left hand rule! There is no actual motion of the wheel in that direction. We just label it that way so we can be consistent and eliminate confusion.
I hope this helps.