The lever, pump, and semi-auto versions of the shotgun were introduced between 1887 and 1900. Why has skeet, trap, sporting clays, and other shotgun sports remained as 1 or two shor games? Why no 4 shot game?
Perhaps this would be better posed at r/Trapshooting, though there don't seem to be a lot of subscribers. But it's fairly straightforward. They are imitating a hunt for game birds, where the hunter will flush one, maybe two at a time, and try to bring them down. It is not unheard of to flush more than two- but then, if you think about it, launching three or four or five clay pigeons into the air at once doesn't grant much time for swinging the gun around and hitting them all before they drop to the ground, regardless of how fast the gun reloads. It gets more into exhibition shooting: Annie Oakley excelled at it ( here's a film of her husband rapidly tossing target balls into the air and her hitting them) but most mortal humans would not.