There are a number of likely reasons for this. Let us examine a few of them.
First, Diamond has not yet started to make his main argument by Chapter 3. In Chapter 1, he sets out a basic factual overview of the state of human technology on the various continents in 11,000 BC. In Chapter 2, he uses the example of Polynesia to show that we should at least consider environmental factors rather than genetics or culture as the cause of inequality between societies. In Chapter 3, he outlines why the Spanish defeated the Inca. None of this is his main argument. Since this is not his main argument, it is not important to rigorously support it with statistics.
Second, it would be hard to use statistics at this point in the book. There have not been very many claims made that could really be backed up by statistics. There are no statistics that can prove that horses helped the Spanish to defeat the Incas. There are no statistics that can prove that writing or centralized government did so. These are claims that seem self-evident, but which cannot really be proven statistically.
Finally, and perhaps most importantly, Diamond needs to use anecdotes for the sake of keeping his readers’ interest. Diamond is writing for a popular audience. He has to use anecdotes to make the book more interesting and accessible. He has to catch his audience’s attention before he starts to make more use of statistics in later chapters.
These are three of the more important reasons why Diamond would not use statistics or other “hard” types of evidence at this point in the book.