Why are loop-induced processes finite without counter terms?

When renormalizing you introduce counterterm diagrams of lower loop order, constructed from counterterm interactions. Importantly, during renormalization you do not introduce any interactions that were not in the Lagrangian in the first place.

So, if there are no tree diagrams to begin with, you also do not have any counterterm diagrams that would contribute at the same order as one-loop diagrams, and you have no way of cancelling potential UV divergences. Your process therefore has to be free of these divergences, since otherwise it would be unphysical.