"Boy Born on Tuesday" a Thornier Problem Than the Monte Hall Problem?

Personally, I find that the ambiguity and phrasing of the question is what leads to pedantic 'gotchas'.

9017 children are born and 4263 are female while the remainder are male; what's the probability that the next child born is female?

It's 50%, if one assumes a bimodal distribution without atypical karyotypic deviations. Preceding births do not determine future births. They are causally independent events.

1245 of those children were born on Friday; what's the probability that the next child is male?

Still 50%. Week day of birth does not impact sex distribution.

It's the way the question is worded that splits this into an intuitive biological problem or an obtuse probability problem.

I have to say, I found your restated problem impossible to understand - I had to google the original problem, which was much more precise and clear:

"I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?"