Yes, I think that everything Grant has been arguing has been correct, except the basic assumption that every number of balls in the urn must be equally likely. Imagine instead that the urns are stuffed using an algorithm that makes it 1% probable that there are 10 balls, and 99% probably that there are 1000 balls. If you pick a 7, what do you conclude is now the probability that the urn contains 1000 balls? The relative probability for 10 balls is .01 times .1, or .001, compared to the relative probability for 1000 balls, which would be .99 times .001, or about .001. This is the same relative probability-- choosing a 7 in this case gives you a 50/50 chance the urn was the 1000 ball or the 10 ball variety! The key point is that to make a meaningful probability argument, you need to know something about how the probabilities are distributed. Probability is about information, it's not something absolute (except in quantum mechanics), and the assumptions you make about the things you don't know (the "rules") are everything.

If this is still unclear, realize that the Carter argument is made from a position of no information, outside of birth order. It is like a person who has just learned how to play chess, entering a chess tournament. This person has no knowledge of how chess tournaments work, other than that there will be 1000 competitors. They don't even know the level of anyone else in the tournament. But they figure, hey, out of 1000, I'll probably end up no worse than 900th place, with 90% confidence. But that's only true if all 1000 competitors also just learned, and are a random cross section from the same population. If instead, this tournament happens to be the World Championships, then it might be quite likely that the newcomer will place dead last, despite their probabilistic argument based on no information. When making a probability argument, the meaningfulness of the result depends on the reliability of the assumptions. Thus the Carter argument is correct as far as it goes, but is of very limited meaning, like the chess entrant's conclusion that he/she will not finish worse than 900th place.