To answer this, it has to be clearly in mind what "the pilgrimage to Canterbury" actually represented and what "a portrait of the nation" actually means. This may be more difficult than it seems. (1) Chaucer's England was not a "nation." It was a culture that was beginning to be unified around the common center of the vicinity of London. Up to and including Chaucer's time, England was an alliance of separate noble powers with each locale having it's own dialect of English and it's own customs and power structure. So Chaucer never represented all of England in a unified way that we are used to thinking about; he represented the emergingly dominant locale of London. [This lack of "national" unity can be seen in the differences in dialect between Chaucer and the Pearl Poet.]
(2) The pilgrims were made up of a divergent types, yet there was one overriding type that interested Chaucer. There was the Miller, the Cook, the Man of Law, a Shipman, a Knight, a Merchant, a Wife, an elected Reeve, among others. These each represent a different economic, social and cultural class. Yet the group that most interested Chaucer and was most heavily--indeed, disproportionately--represented was those in religious orders. Ten of the pilgrims' tales are told by individuals connected in one way or another to the Church and serve in some sort of religious ministerial capacity. So while Chaucer gives a broad image of a cultural cross-section, he nonetheless particularizes the cultural image to a specific segment that he is especially satirizing and exposing, that being the corrupt and hypocritical religious community.
So now, in order to find a parallel in our society, we must ask: What unified cultural condition did Chaucer's Tales represent? While the pilgrims in the Tales represented divergent economic and social levels in a disunified cultural association, the pilgrims also significantly represented a governing body that was most corrupt and deceptive in its leadership. Now the question is, what, if anything can bring these elements (divergent socio-economic levels and decisively corrupt governing leadership) together in American culture?
First: I think of sporting events that do draw many levels of socio-economic representatives together (though not all for tickets to sports events are a luxury items and not for all in many regards). Second: I think of two possible venues. One venue would be a large and popular sports event held in Washington, D.C. since most socio-economic levels would be represented and since (what many think of as corrupt) government leaders would join in and be in the same audience. The second venue would be Annapolis, Maryland, during the annual Army-Navy football game. The same scenario applies here: most socio-economic levels and (what some consider) a corrupt leadership body would be gathered in one place, bearing in mind that many think the military hierarchy is corrupt.