I agree with you that "I know any kid in the school" sounds quite odd. I do not believe, however, that the oddity derives from the verb "know." Indeed, "I recognize any mushroom in the forest" sounds odd to me as well.
I think the issue comes from the meaning of "any." According to Merriam-Webster, "any" in the sense of a determiner means "one or some ... of a number of things" or else "whichever of a specified class may be chosen."
Now in the sentence, "I know any kid in the school," the second meaning works only if you know every kid in the school, in which case you must know more than just some of them. Under that supposition, it is a lot clearer and so more idiomatic to say "I know every kid in the school" or "I know all the kids in the school." Moreover, if the school is small enough, it is credible that you may know them all.
The examples in which the use of "any" do not sound odd to me include "can." Yes, the sentence "I can drive any car" logically implies the sentence "I can drive every car existing on earth," but it is not seriously contemplated that you will ever have the opportunity to do so. What is really being contemplated is that you can drive that relatively small number of cars that you have an opportunity to drive.
In short, when "any" necessarily implies "all" and "every" and "all" is plausible in practice, "any" will not be idiomatic. It is when "all" is not plausible in practice, that "any" becomes an idiomatic choice.
I must admit that I have not found a supporting source for this explanation of what your own ear has already told you (though I shall look at Fowler when I get home). Instead, I have relied on my own sense of which word I would employ in different circumstances.