in edited, expository prose, you (usually) want to be concise
I like concision in the written word but I also agree with languagehat: redundancy and lack of concision is part of the vernacular and “baby boy” doesn’t annoy me one bit because I think “baby” implies softness, sweetness, or whatever qualities you like to ascribe to young life. You could take the analysis to the other extreme and argue that “gave birth to a boy” actually does imply she gave birth to a male child of indeterminate age so needs the qualifying “baby”.
I also said the following:
The problem with the omit needless words dictum is not the principle, but that thoughtless writers and editors apply it too stringently and without regard to context. They trim away words that really are necessary or that may not be strictly necessary to get the message across but add character and flavor to the prose.
In this case, it is a news report, where one would expect concision and an objective relating of facts without embellishment. In other contexts, a familial birth announcement or fiction for example, you can add the alliteration or qualities of youth as you wish. I also said: “it’s very far from the worst thing you could write.” I wasn’t saying I would object to baby boy, just identifying it as unnecessarily redundant.
But I would strongly disagree with any analysis that said using an unmarked boy in give birth to a boy implies an indeterminate age. Giving birth is the marker that indicates infancy.