Yes, the bolded portion is a participle phrase or clause (depending on your grammatical sect).
And Yes, the clause is supplementary to the main clause.
But you cannot "turn it back to an adverbial clause" because it does not act as an adverbial—that is, it does not modify a constituent which acts as a verb, adjective or adverb. It modifies a noun phrase, the song, and thus acts as an adjectival.
To turn this into a full dependent clause you would employ a relativizing "conjunction", which would require placing at after the nominal it modifies:
The song, which was first introduced in the 1940 Disney movie Pinocchio, tells us &c
Or you could, by appropriating its subject from the main clause and providing a copula, promote it to a full independent clause juxtaposed to the main clause or joined to the main clause with a "co-ordinating conjunction such as and or a semicolon, which would require filling the subject slot of the original main clause:
The song was first introduced in the 1940 Disney movie Pinocchio; it tells us &c