With the failure of the Volcanic to point to, it's easy to see why revolvers stayed on top for as long as they did, but when Winchester repeaters started taking over the single load rifles it's a little confusing to wonder why a similar design didn't take off for handguns. Why didn't a repeater or "lever action" handgun design ever become prominent between the Winchester rifle's claim to fame and the auto loading handgun becoming popular?
Why would they? The advantage of a lever action over a revolver action in a rifle is a larger magazine under the barrel of the rifle, compared to the 5-7 rounds in a revolver cylinder. With the short barrel of a pistol, there would not be a significant increase in the number of rounds. Second, with a lever action rifle, the weapon may be held with the rifle held pointing down range in the left hand with the butt against the shoulder and the lever worked with the right hand. In contrast with a lever action pistol the butt would be held in the right hand and the action worked with the left. This would result in the weapon not being pointed down range and potentially (likely) pointed down the firing line. The safety implications should be obvious. Even a single action revolver would be back in battery faster, once double actions came along all the more so. So revolvers would be faster and safer and have the same number of rounds. Other than (potentially) muzzle velocity, what is the advantage of the lever action pistol?
Its also worth noting that with a rifle you have three points of contact your shoulder your off hand and the hand in the middle so you can keep and rifle very stable while cycling the action with a pistol it will be nowhere near as stable and as such it will be slower
Source have used a a lever action rifle and a mares leg (lever action with a short barrel and no stock)