It was interesting for me to look that way at my little list -- the list certainly isn't all the songs I know, but what I'd want handy if called upon at a session... thus no "Deck the Halls", and probably also the reason for the dearth of religious items. Shows up, also that I don't keep too many chorus songs on that list, it's more about performance (for myself or others) than group singing.
Any act of categorization has its uses in shaping an analysis or in providing a framework for practical application. But, it seems to me that categorization (if not categorizors!) is always going to be open to alternative approaches. (And haven't we found that to be true here?!)
Not having read the book (and this being true for everyone commenting here), it's hard to form a cogent argument one way or another about this particular framework of categories. It sounds to me like it was an interesting exercise for looking at how song has operated as a form of communication and record-keeping. To me, that overarching theme is more interesting than arguing about particular categories.