Sure, and if the quibble was along the lines of "You never explicitly said boys and girls are 50-50 distributed! You never explicitly said elder and younger childrens' birth genders are independent! You never explicitly said birth-days-of-the-week are uniformly…", then that would be fair, if pedantic.
But this "You know what Mr. Jones has told you, but you don't know under what circumstances he would have told you this" objection is objecting to some other problem than the one posed; the problem posed had nothing to do with Mr. Jones saying anything.
I understand the reason for worrying about this, because many probability riddles ARE poorly worded or presented in such a way as that this becomes an issue, but it wasn't the case here. (Note: I haven't watched rest of the video and have no comment on it; I'm just considering the wording of this individual question within it)
There was never any claim that Mr. Jones said anything, and no one was called to infer anything from any actions taken by Mr. Jones. He could be a lifelong mute. Rather, the fact that Mr. Jones has two children was presented, by an omniscient narrator, and then a counting question was asked.
(Indeed, Mr. Jones himself is completely irrelevant to the problem asked, except as a way of framing the counting question to be about two-children families. The question asked might as well have been "What proportion of two-children families with a boy born on Tuesday have girls?". It was very slightly differently worded, but not in such a way as makes "We don't know what Mr. Jones was asked!" a relevant objection.)
