Hence, the original question: how come "write" is a strong verb, retaining IE forms, when IE was not, as far as we know, a written language at all? Well, as it turns out, "write" only goes back to Old English, where it seems to have carried the idea of tear or scratch. It carries what I think of as IE forms because OE largely preserved such forms. Only the Gc languages share this word as meaning "scratch," and even then, only OE had the special sense of scratching on bark (or perhaps to form words. (The bark, incidentally, was probably beech, from which we get "book." Cognates for "book" are found in the other Gc languages, which argues against any inference that might be drawn from "write" not having cognates with a literate meaning.))
It's common in many languages for a word which originally meant "scratch" to come to mean "write." It happened in Greek, from which we inherit many words containing -graph, which was Greek for scratch > draw > write. The same IE root also gives us "grave," which is different kind of scratch, and "graven." A word for scratch also became the word for write in Latin. This is where English gets script, inscribe, and similar words.
Speaking of books, do they have "leaves" because book means beech? The IE root is *leup-, meaning to peel off, break off. This sense was applied to tree bark from early times, generating a Gc word for roof (made from bark), whence "lodge," "lobby" (originally a monastic cloister), and other words applying to roof-like objects. So, the answer is yes: since one form of early books were bound or bundled sheets of bark, the association with "leaf" was an early one. In fact, our special sense of leaf meaning the green petals of tree apparently developed later.