I've investigated my own question a bit and I believe I can answer it fully now.
Mattino, mattina and pomeriggio are nouns and not adverbs. They don't have a double noun/adverb form like oggi or domani (I can say "Ci vediamo domani", but also "penso al domani").
When I say "Ci vediamo di mattina" or "Ci vediamo domani pomeriggio", "di mattina" and "domani pomeriggio" form a "complemento avverbiale di tempo". However this is logical analysis which is not what the question is about.
Grammatically speaking, "mattina" and "pomeriggio" remain nouns, and that can be seen in the (relatively archaic) usage of the words in the form "ci vediamo la mattina" or "Ci siamo incontrati un pomeriggio".
Nouns take articles, adverbs don't. For the same reason "ci vediamo pomeriggio" is incorrect, as the lack of an article identifies "pomeriggio" as an adverb, but there is no such form of the word.