Regarding the sentence *A unified exposition of their theory, however, is lacking in the literature,* I think the theory of [combinatorial species](http://en.wikipedia.org/wiki/Combinatorial_species) was invented towards this aim.  Indeed, combinatorial species were invented by Joyal in 1981 (9 years after *On the foundations of combinatorial theory. VI.*).

Regarding (3), I also have no idea what it means.  I will however throw out the possibility that this could be another instance of Gowers's [two cultures](https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf) phenomenon in mathematics.  That is, there are the people who use generating functions as mainly a tool to solve problems (I would classify myself into this group).  And there are others who now study them as a theory in their own right, but those people didn't exist in 1972.  Of course, as Gowers warns, the division line is often not so clear cut, but this could be what Doubilet, Rota, and Stanley were alluding to.