Where you're getting confused is by trying to combine state space determination and probability determination at the same time (this is also why the problem is so similar to Monty Hall). The state space is shifting when you say "assume X, then the probability of Y." You are going back and forth between using and not using the information to decide arbitrarily that some probabilities are 50% and others are 0%, which leads to an invalid conclusion.

Specifically: it is not true that the firstborn has a 50-50 chance of being a girl, given you were told that the family has at least one girl. The firstborn has a 2/3rds chance of being a girl. This is the heart of your confusion.

In a broader sense there is an entire class of confusing conditional probability problems like this. Events which are causally independent in reality (e.g. gender of a child, which door Monty Hall hid the car behind) fail to be probabilistically independent when you have extra information. Yet these probability games are contrived in a way that our intuition takes over and we use our causal understanding even when a better probabilistic understanding gives you a better answer.

But in the monty hall, the "discloser" is limited by my choice/observation. It is literally part of the framing. In this framing, there is no limit based on my choice.

Consider, if you tell me that the Smiths have at least 1 girl, and I meet their daughter but you haven't met either, I have no way of knowing if I met the 1 girl or not. I could ask you, but you would just say, "I don't know, could be her. Could be the other kid. I just know they have at least 1 girl."

This is very different from the monty hall case, where the announcer knows what is behind all doors.

Similar. But different. I was using coins as an example elsewhere. If I flip a dime and a quarter, and tell you that one of them is heads, do you have increased chance of knowing if either particular one is heads? This is more liar's dice than it is monty hall.