They saw some use throughout the 1860s, but never really caught on. It's estimated that, in all, Colt made about 1,100 of these revolving shotguns. They did make a 20-gauge version available, but that was clearly more for bird hunting, and not appropriate for social work.
Postscript: One of the hazards associated with black powder revolvers was that a bad ignition could fire off all cylinders at the same time. Picture that happening with this... (Or try not to.)
And: They're around today. One in "good" condition will cost you about eight thousand dollars; in "fine" condition, closer to ten.