First, let us note that Diamond does not say all societies pass through all 4 stages. Some societies do not progress past one of the lower levels. However, he does claim that every society that gets to a higher level does so by passing through the previous levels.
Diamond can say this about every society because he is drawing on social scientific and historical literature that documents the various types of societies that exist today and that have existed. By looking at this literature, he can note that societies generally fall into these four categories. We should note that Diamond himself talks about how these categories/stages are imprecise and are "doomed to imperfection (p. 267 in the paperback edition)." Diamond can then look at the fact that some societies have moved from one category or stage to another. He can then extrapolate from these facts to infer that all societies that got to be advanced did so by moving through these stages.