I think the explanation is wrong. It is based on the probability of having combinations of boys and girls and then counting the combinations that have at least one girl, but this is not the situation: there is no probability in question for one of the kids, it is a confirmed, past event and the only other probability is the sex of the other kid.

Otherwise you can derive any probability as a branch of a probability tree that contains it and calculate the probabilities of the tree and then the one of the branch. This makes no sense.

For example, a family has a kid and the kid is a girl. The family wants another kid; what is the probability to be a girl? Is it 1/4 because having 2 girls is 1/4? No, it is 1/2 as it is for any new kid.

Your example is different in a critical way. These two questions are not equivalent.

Q1: I looked at only one of a pair of two randomly selected children and it was a girl. What is the probability the other I didn’t see is a girl?

Q2: I looked at both of two randomly selected children and at least one of the pair of children is a girl. What is the probability the other is also a girl?

The question in the article is the second question, not the first. The fact that the observer looked at both children and not just one of them is crucial. As is often the case in these puzzles the exact information available is the critical issue.