I can understand how you'd get that confusion from the names of the approaches, but you've got it rather wrong. Bayesians incorporate all evidence that frequentists use, including observed frequencies in large data sets; this results in their posterior distributions being centered very narrowly on the frequentists' point estimate.

The other branch that I don't remember the name of assumes that all probabilities are equal i.e. P(A) = P(B) = P(C) = ...

Uh... is that a real thing?