Earlier quoted context omitted.
It's more like seeing one boy actually impacts the gender of the second child, which is why the quantum experiment is so mind boggling. It doesn't seem to have any resemblance with classical physics and only makes sense in quantum physics.
That's exactly what the paradox states. Seeing one boy does affect the gender of the second child.
Here's another example: If I flipped 100 coins, and told you there were 99 heads, what's the likelihood the other 1 was tails? The answer is 100/101, because there are 100 ways the one tail could happen, while only one way zero tails can happen. But if you're only told that the first 99 are heads, the answer is 1/2. You know the last flip has exactly the same odds as all the other ones -- the trials are independent. The difference is in one all the flips were included in the results (and the question isn't about any one specific flip), while in the other you had no information about the last flip (which the question was about).
Another way of looking at it: In the first example, the question was asking about the sum of 100 trials and you were given information about 99 of those 100 trials. You were given 99/100s of the information you need to answer the question. In the second example, the question was asking about the last trial and you were given information about 99 unrelated trials beforehand. You were given none of the information you need to answer the question.