Nevertheless, which event with a symmetrical alternative were you referring to?

Given that the women does have a boy and a girl, what is the probability that she would state that at least one of them is a boy? By symmetry, you would expect a priori, not knowing anything about this person's preferences, that in the same conditions, she is equally likely to state that at least one of her children is a girl, 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. (The case could be made that the statement is also evidence that the woman has a tendency to say at least once child is a boy rather than that at least one child is a girl. But this is a small effect, and still does not justify assigning a conditional probability of 1.)

I think the frequentist approach could handle this problem if applied correctly, but 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.

any good textbook will always define exactly what the context is so that there is no guessing and no disagreement.

If you read the article I cited, Eliezer did explain that this was a mangling of the original problem, in which the mathematician made the statement in response to a direct question, so one could reasonably approximate that she would make the statement exactly when it is true.

However, life does not always present us with neat textbook problems. Sometimes, the conditional probabilities are hard to figure out. I prefer the approach that says figure them out anyways to the one that glosses over their importance.