It seems like you're choosing Avenue Q, Billy Elliot & Book of Mormon because of the foul language & in the case of the latter two, some crude humor. I don't know if that spells "controversial" to me other than perhaps "unorthodox" from typical Broadway fare.
I think Scottsboro Boys is a better example, although yes, it seems like the real controversy stemmed from the protests by the people who totally misunderstood the show. But it was a bold show. And I loved it. Too bad the bad press kind of killed it.
I never saw the original production (only the stripped down staging at the Delocorte) but The Capeman comes to mind when thinking of controversy. I remember reading about family members of the Capeman's murder victims protesting outside the theater on opening night. They thought the show was glorifying the killer. Seems pretty controversial to me.
Rent is arguably controversial. I don't really find A Chorus Line controversial though. Groundbreaking, perhaps, but not really controversial.
I'm too young to dip back into the 60s/70s/80s but it seems like there were probably lots of radical, lesser known shows that would have proved controversial.
Interesting topic though. Have fun with it.