Eliot originally intended to title the poem He do the Police in Different Voices. This was an allusion to Charles Dickens' Our Mutual Friend where Mrs. Betty Higden describes how well Sloppy reads the newspaper aloud.
This site has an excerpt from that section of Our Mutual Friend (taken from Book the First, Chapter 16, "Minders and Re-minders").
The draft title of He do the Police in Different Voices was an allusion to help the reader to understand that, while there are many different voices (speakers) in the poem, there is one central consciousness. After changing the title Eliot may have felt it necessary to add his note about Tiresias:Tiresias, although a mere spectator and not indeed a 'character', is yet the most important personage in the poem, uniting all the rest.
Citation: "The Facsimile" pp. 4-5, 125