I think R2-D2's role in Star Wars is that he represents the audience themselves, this (mostly) quiet little observer of all of these grand events from literally the first Act of Episode I to the final battle of Episode IX.

And when R2 does "speak," he speaks for the audience as well. Every excited little beep, whine, or groan reflects what the audience is feeling during any given scene, from excitement to sadness to irritation.

In some ways, R2 is the chronicler of all the events of Star Wars. Thus it was no surprise at all to see R2 make an unexpected appearance in the most pivotal scene of the Mandalorian's intersection with the rest of the Star Wars universe: the ending of one story and the beginning of another.