Is the curl of a B field at a point on a wire infinity?
From the mathematics alone, yes, this does go to infinity.

But notice that this assumes an infinitesimally thin wire with infinite current density; while the approximation of a thin wire is often useful, it breaks down, as you noticed, when you start dealing with magnetism in anything more than a cursory manner.

If the wire has finite size, the integration contours that are located inside the wire also have less current enclosed, so the singularity disappears.