so to assign the conditional probability higher than .5 does not make sense, so it is definitely not right for the frequentist Eliezer was talking with to act as though the conditional probability were 1

In the "correct" formulation of the problem (the one in which the correct answer is 1/3), the frequentist tells us what the mother said as a given assumption; considering the prior <1 probability of this is rendered irrelevant because we are now working in the subset of probability space where she said that.

it seems that frequentist in practice get it wrong because they do not even consider the conditional probability that they would observe a piece of evidence if a theory they are considering is true.

Considering whether a theory is true is science -- I completely agree science has important, necessary Bayesian elements.