With objective priors one can always ask “so what?” If it’s not my subjective prior, then its posterior will not equal my subjective posterior. There isn’t an obvious way to bound the difference between my subjective prior and the objective prior.
With frequentist methods it’s possible to get guarantees like “no matter what prior over θ you start with, if you run this method, you’ll correctly estimate θ to within ϵ with at least 1−δ probability”. It’s clear that a subjective Bayesian (with imperfect knowledge of their prior) might care about this sort of guarantee.
With objective priors one can always ask “so what?” If it’s not my subjective prior, then its posterior will not equal my subjective posterior. There isn’t an obvious way to bound the difference between my subjective prior and the objective prior.
With frequentist methods it’s possible to get guarantees like “no matter what prior over θ you start with, if you run this method, you’ll correctly estimate θ to within ϵ with at least 1−δ probability”. It’s clear that a subjective Bayesian (with imperfect knowledge of their prior) might care about this sort of guarantee.