"But now the number of fingers you held up doesn't equal the number of items you were counting."

That's true in either case though. Once you finger overflow the real representation is up to interpretation of what came before and what the known base is. That's why (as stated in a previous response) I think base 6 is most appropriate for our hands. You can accurately and naturally represent numbers 0 to 35 with no weird shenannigans.