That sampling arises because being part of the population which has no boys is * necessary and sufficient* to (truthfully) make the statement that forms the paradox.
The alternative interpretation of the paradox arises when the wording of the paradox is construed to identify one of the children as male or female. In this case (stating something like "my first child is male"), being part of the population (x \in {BB, BG}) is necessary and sufficient and leads to the 1/2 probability of having two boys.
In short, the question becomes whether you believe the child is identified in the wording of the question. Honestly, the author of the paradox goes pretty far out of their way to say "at least one of the children is male" avoiding that identification.
The alternative interpretation of the paradox arises when the wording of the paradox is construed to identify one of the children as male or female. In this case (stating something like "my first child is male"), being part of the population (x \in {BB, BG}) is necessary and sufficient and leads to the 1/2 probability of having two boys.
In short, the question becomes whether you believe the child is identified in the wording of the question. Honestly, the author of the paradox goes pretty far out of their way to say "at least one of the children is male" avoiding that identification.