There are two main reasons for this.
First, writing does not really have “overwhelming value” for everyone. Writing is, for example, essentially useless for a hunter-gatherer society. It is only very useful for highly developed civilizations that need to do things like keeping track of edicts by the kings and of who owes how much in taxes. Writing is very hard to develop. If you have a society in which writing will not do you much good, there is no point in developing it.
Second, it is very hard to develop writing and so most societies only get writing by borrowing it. As Diamond says (on p. 237 of the paperback edition of the book)
…the vast majority of societies with writing acquired it by borrowing it from neighbors or by being inspired by them to develop it, rather than by independently inventing it themselves.
This means that peoples who live in places where someone has already invented writing are much more likely to be able to get writing systems of their own. Many peoples did not live near enough to a place that developed writing to be able to borrow it. As Diamond says (again on p. 237)
Had they been located nearer to Sumer, Mexico, and China, they might instead have acquired writing or the idea of writing from those centers…
Thus, we can see that peoples did not develop writing because it was not useful or because they did not have the chance to borrow it from others.