Can a noun function as an adverbial?

English doesn't like to use prepositions before words like last, next, this or every. It is evident when you contrast

I go to church on every Sunday.

with

I go to church every Sunday.

There is no reason to use the preposition "on" in the above example. This Ngram Viewer shows a declining trend of using "on every Sunday"

The reason why Sundays is listed as an adverb (in some dictionaries) is it is very broadly used in place of "on Sundays" and it functions as if it were an adverb.

However, I don't think these nouns are adverbs. They are adjuncts and they are used to complement prepositions. Even if a preposition is omitted, there is no change in their meanings.

Yes, no problem:

They leave Sunday.

Sunday is a temporal adjunct here (read adverbial). It is a noun phrase consisting of a single noun.