If you browse YouTube looking for Cowboy Action clips, you'll see that CAS shooters have really perfected running these "old guns" pretty darn fast. After shooting CAS for 5 years, I would never feel undergunned with a lever gun or double/pump shotgun. Even us mere mortals (e.g., not Grim Reaper, Lead Dispenser, etc) can put out quite a bit of lead with these.
I know I've told this story before, but my old club used to sponsor a "Cops vs. Cowboys" charity shoot. Cops got to bring their duty guns...semi auto pistols, pistol caliber rifles (several 9mm ARs, UMPs, MP5s on semi) and semi-auto/modern pump shotguns...against our single action revolvers, lever guns and double guns/Win97 pumps. The only restriction is we downloaded their guns to match our capacities.
Guess who won regularly? If you say the cops, you would be wrong. Never saw such a bunch of bad gun handling as when the cops shot.