Hi Sgofpl,
I can't illucidate but I too am confused. I've often seen these analyses offered, seemingly as options, where something is said to modify the whole phrase or clause rather than a particular word. I'm looking forward to some learned opinions.
Your use of "than" is not typical. Do you mean, "rather than" or "first A, then B"?
My only comment on the example itself is that once you start playing this game, you can say that a whole phrase or clause is acting as a noun, so its modifiers would act as adjectives. But I really get the sense from this example that "adding" modifies everything which comes before it in the sentence.
Regards, - A.