Indexing is not for counting elements, it's to reach a particular one.
I still find it more useful to name it my tenth finger, rather than the ninth from the first. And I'm fine with my middle finger, finger number three - being the one that sits between the pairs of fingers made up of finger number one and two, and fingers four and five. I don't really see how it's any better to count up from the zeroeth offset of the first finger to the second offset to get to the middle finger.
I see it as rather simple. If you're listing off-sets, starting at 0 makes sense, because there'll be something there. But if you're counting elements starting at 1 makes more sense. I don't have 9 fingers offset from my first one, which I call finger zero; I have ten fingers, starting with the first one.
A set with no members is the zero set, once you add one element, you have a set of one. An empty list contains no elements. It does not have a first member, because it's empty. Add an element, and it will have one member, and it will be in the first position. Not in the zeroeth offset from the beginning.
You can the numeric names of letters of a string in an array, but I don't see that as a compelling reason for naming the first letter of a word, the letter which is at the zero-eth offset from where the word is held in memory. At best, it's a low level optimization, at worst it's just a leaky abstraction. Maybe you want to store the length of the word there (like in Pascal), for a different trade-off in terms of what is optimized.