I give you some numbers taken from a normal distribution with unknown mean and variance. If you're a frequentist, your honest estimate of the mean will be the sample mean. If you're a Bayesian, it will be some number off to the side, depending on whatever bullshit prior you managed to glean from my words above - and you don't have the option of skipping that step, and don't have the option of devising a prior that will always exactly match the frequentist conclusion because math doesn't allow it in the general case . (I kinda equivocate on "honest estimate", but refusing to ever give point estimates doesn't speak well of a mathematician anyway.) So nah, Bayesianism depends on priors more, not "just as much".

A Bayesian does not have the option of 'just skipping that step' and choosing to accept whichever prior was mandated by Fisher (or whichever other statistitian created or insisted upon the use of the particular tool in question). It does not follow that the Bayesian is relying on 'Bullshit' more than the frequentist. In fact, when I use the label 'bullshit' I usually mean 'the use of authority or social power mechanisms in lieu of or in direct defiance of reason'. I obviously apply 'bullshit prior' to the frequentist option in this case.