"For example, why do we usually label δέ “particle” and καί “conjunction,” when the connective functions of the former and the adverbial ones of the latter are amply exploited in our source texts?" One answer to this question is that the particles involve presuppositions and the conjunctions involve a syntactic pairing relationship. In other words, the "connective" functions that you are describing for the particle and the conjunction is similar in a way, but it results from distinct semantic/pragmatic mechanisms.