Just because he wrote the Cambridge Grammar of the English Language doesn’t mean he’s right about what a preposition is.
I think a more useful way to approach this is that he uses a different definition of “preposition” than do you. The discussion which follows this approach is regarding what definition is most useful.
This is similar to the future tense discussion, which is really about the definition of “tense”. I am persuaded that the traditional definition applied to English grammar is a hopeless mess, conflating syntax and semantics is generally being poorly thought out. I think an argument can be made for abandoning the word “tense”, as it causes confusion, but that is a slightly different matter.
I haven’t studied the preposition issue enough to have an opinion. In the meantime, I just keep in mind that different people use the word “preposition” differently, and try to keep straight who is using which.
