*Now, if at least one child is a boy, it must be either the oldest child who is a boy, or the youngest child who is a boy. So how can the answer in the first case be different from the answer in the latter two?*

Because they obviously aren't exclusive cases. I simply don't see mathematically why it's a paradox, so I don't see what this has to do with thinking that "probabilities are a property of things."

The "paradox" is that people want to compare it to a different problem, the problem where the cards are ordered. In that case, if you ask "Is your first card an ace," "Is your first card the ace of hearts," or "Is your first card the ace of spades," then there is the same probability of 1/3 in all three cases that both cards are aces given an answer "Yes." In that case the averaging makes sense because the cases are exclusive. In the "paradox," you can't average by saying that, "well, if there's one it's either the Ace of Spades or the Ace of Hearts, and in either case the answer would be 1/3, so it averages to 1/3." The problem is that you're double-counting.

I'm a Bayesian, but I don't see what this particular example has to do with subjectivity and agents. Probability is a result of the measure and the universe one is dealing with, and that may lead to results that seem unintuitive to those who don't grasp the mathematical principles (that seem obvious to me), but that has nothing to do with needing an agent. Define the measure space as you have done, claim that the probabilities are cold hard inherent facts about the objects themselves, and the result is independent of an agent.

This "paradox" seems on the same level to me as the confusion as to why the chances of rolling a 6 in three rolls of a die is not 1/2, or the problem that if one takes an outbound trip averaging 30 mph, then it is impossible to make the inbound trip so as to average 60 mph without teleporting instantaneously.

Or, I suppose, I would compare it to the other noted statistical paradox, whereby a famous hospital has a better survival rate for both mild and severe cases of a disease than a less-noted hospital, but a worse overall survival rate because it sees more of the worst cases. Merely because people don't understand how to do averages has little to do with them requiring an agent.