Why is radiometric dating only considered a use of alpha and beta radiation?

Radiometric dating tends to use a nucleus that changes into some other easily distinguishable nucleus. For example uranium decays to lead 206 and 207, which can be easily measured in a mass spectrometer. We measure both the uranium concentration and the lead concentration and infer the age from how much of the uranium has changed into lead.

The problem with gamma radiation is it doesn't produce a chemically distinguishable product. Gamma decay is effectively a decay of the excited state of a nucleus to a lower energy state of the same nucleus. So there is no way to tell how much of the original parent nucleide has decayed.

By contrast alpha decay produces a daughter atom with an atomic number lower than the parent by two, and beta decay produces a daughter atom with an atomic number higher than the parent by one. In both cases a mass spectrometer can easily tell the difference between the original atom and the product.

John's answer is only part of it. There are gamma emitters that aren't simply due to an excited nucleus. However, the ones that occur naturally are part of the uranium decay sequence. They are not at either end of the chain and thus measuring the quantity relative to their parents or daughters only tells you the ratio of the half lives, almost nothing about the age of the item.