So there's "iconic" and there are "the best showcase of KC's BBQ", sometimes those two things intersect but not always.
I really like Q39, SLAP's, Jack Stack (closer to the "traditional" end of the KC spectrum), Char Bar, and Joe's KC. Char Bar has a nice outdoor area with a bar, yard games, and dining seating...might be a good place to make an evening of. Gates and Arthur Bryant's are some of the more prominent "old school" KC 'que joints, of those two I prefer the latter. Zarda's, and Smokehouse are pretty good. LC's is another place that is probably considered "iconic" as far as KC barbecue is concerned, I believe I read that the owner/founder recently passed away. Woodyard BBQ is pretty good, as the name suggests, they sell wood there in the parking lot area next to the restaurant. Harp BBQ has gotten a lot of attention recently, although I've not had it...I think it's a "Saturdays" and or "popup" kind of thing. He operates out of Crane Brewery on Saturdays until sell out, and their beer is pretty good too.
There's a lot of good places to have barbecue here, I certainly haven't listed them all. Just kind of depends where you're staying and how long you'll be here.