Why do you think a removable discontinuity (hole) doesn't produce an asymptote on the graph of a polynomial function, even though it is excluded from the domain of the function?

Because  removable discontituity means that the limit of the function at that point has a finite value, and then you define the value of the function as that valu (the limit value).

An asymptote means that the limit of the function goes to positive or negative infinity.

You cannot meet both conditions, finite  and infinity limit at the same time.

Is there a picture?