Elaborations of cotrs and calfbirds in US:
Guianan cock-of-the-rock: Dallas World Aquarium has recently begun having great success with the species, and they have about 30 behind the scenes (none on exhibit). San Diego keeps 2.2 on exhibit, no breeding success.
Calfbird: the first breeding success was at the San Diego Zoo in the 1990s. That was all, until DWA hatched some in 2011 and now has about 20 on and off exhibit. After that breeding, the program succeed more with a number of hatches each year. This year, SDZ had success for the first time since those original successes with their on-exhibit pair. Bronx and Cincinnati are also holders, with a 1.1 pair and 0.1, respectively.
Andean cock-of-the-rock: DWA is of course the primary proprietor of this species in captivity, breeding them quite well and with over 50 specimens on and off exhibit. There are seven other American holders, all with significantly lower numbers, though. I believe San Diego has recently begun having success, and Bronx is attempting some different strategies to get their pair to breed.
Hope I helped!
Edit: I noticed you also mentioned manakins. A species that DWA doesn't have is the white-bearded manakin. San Diego Zoo is the only holder (as far as I'm aware). They hatched the first two specimens in America in 1998, though today, all that remains is one old female that is off exhibit.