If I understand your argument, you think that:
(1) there is no way an acquaintance would remember your child's birthday organically,
(2) so if they do remember they have taken actions to remember,
(3) and the reason they took those actions was in order to give the appearance of caring about the relationship between you both.
(4) Furthermore, everybody on the receiving end of this interaction finds it distasteful or offensive, or at least will not think more highly of you because of it.
I think that (some of) these points may hold for some people, but don't hold in general.
1. For whatever reason, there are people who will remember things like this.
2. There are also people who may remember something like this in one situation, but not in general. For example, it just so happens that my wife's brother has the same birthday! We went out for his birthday last night, which prompts me to remember this detail about your child. So when I meet you soonafter, I ask "how did <eldest child> enjoy their birthday?"
3. The important part of this is the 'appearance of caring', I think. Is it weird for a presenter at a workshop to use memorisation techniques to remember the participants names? Is it weird to make flash cards about people you met from a networking event to memorise who they are and what they do? There is a really blurry line between the 'appearance of caring' and 'actually caring' and I'm not sure there is much of a meaningful distinction.
4. I'm part of a large family conversation online. Many of the posts are birthday wishes, often from people you might see once a year. The first post is often from one aunt in particular, who has obviously collated a list of the birthdays for the ~50 people in the chat. No-one thinks this is creepy, and everyone loves that she has gone to the extra effort to remember their birthdays. Posting birthday wishes to an old college acquaintance because facebook reminded you isn't creepy, but also isn't viewed as fondly becuase less effort is required. I think it's reasonable to imagine many people would be flattered that someone went to the trouble to remember personal details.
I hear your point though, and agree that it could be creepy - it really is about understanding your audience, as you say, however I think it is more likely to be appreciated than not.