Residue Theorem if Pole is on Contour

Very reasonable question! I wondered about this for decades, myself! :)

The integral does not literally converge. It does converge in a "Cauchy principal value" sense, but this requires that we make a convention, or do something. It is not in any way automatic, any more than $\int_{\mathbb R} f(x)/x\;dx$ "automatically" takes the "Cauchy principal value" value.

The more bare, real fact is that that integral "through" the pole is not well-defined, since, after all, as a literal integral (as opposed to something with conventions imposed) it does not converge at all.

This explains why there's no "proof" that a contour integral "through" a pole picks up half the residue. Because the assertion is not literally true, as stated. Sure, we can say something about the related principal value integral, but that is a very different thing.

(And the possibilities of other "angles" of contour through poles likewise need principal value interpretations, otherwise are not well-defined. And, NB, there is no mandate to take the PV interpretation, so, in particular, the literal integrals do not magically/automatically take those values.)

EDIT: also, in case people might too glibly assume that there's not real issue about "regularizing" such integrals, please do consider the precise assertion of the Sokhotski-Plemelj theorem (eminently google-able). That is, it turns out that it is easy to imagine false things in terms of regularization.