It seems to me like you already answered your own question.
"What is to be done is to add up both magnetic fields by superposition and therefore we indeed have the sum of the magnetic filed of the hole being ZERO, plus the magnetic field of the cable which is NOT zero at that point."
You are interested in the sum of individual contributions. Think about if you had a wire with current flow and were trying to determine the resultant magnetic field. You could have any number of ampere loops in which the wire is not enclosed and their resulting contribution is still zilch.
In your problem with the ampere loop from r = 0 to r = b is telling you that there is no contribution to the magnetic field from this portion of space since there is no current flow there.