Why use gamma over alpha radiation?

Gamma radiation is used when the radiation source is outside the body and we need to focus it into a tumor that's inside it. For these situations, if we used alpha radiation, it would just get stopped at the skin, which is definitely not a good thing.

This type of external-beam therapy can also be done with charged particles, known as particle therapy, in which case you have the advantage that the sources can be more consistent and that you have better control over the focusing (since you can use electrostatic lenses and magnetic fields to shape the beam). However, once you're in that arena, proton therapy is likely to have every advantage of helium-ion beams, and it will be much easier to produce.

Alpha emitters are good in situations where you can get them right next to the tumor cells you want to kill, which probably means that you're including the alpha emitter in some biochemically-active molecule (a radiopharmaceutical) that gets preferentially concentrated in the tumor.

This does seem to be used in practice, though it seems that most therapies of this type use beta emitters, which have a slightly larger radius of action.

Alpha particles are absorbed too easily; usually within a couple of centimeters. Gammas have no such issue. Protons, on the other hand are excellent for radiation therapy because their energy can be tailored to produce a "Bragg peak" (see Wikipedia) at a selected depth, and they stop there. Any ion (protons are hydrogen ions, alphas are helium ions) shows a Bragg peak.