It's right there in the name. Auto
5.
5 shots. Have you ever hunted fast birds? Shooting a pump gun fast and accurately while swinging is a skill that some have developed, but not everyone wants to. I don't, for example. People wanted a gun that would fire 5 shots as quickly and conveniently, while swinging on a target, as a double gun will shoot 2.
Now, in many places, and for waterfowl, doves, etc., you can only use a 3-shot gun. With only 3 shots, I'd probably rather have two chokes instead, or a pump gun can help conserve ammo and encourages careful shots. But back when the Auto-5 was introduced, you could use it just about anywhere, legally, at full capacity.
Here in Idaho, we can hunt non-migratory upland birds without a plug. Last Fall, I was out hunting. Someone else flushed out a pheasant, shot and missed. It came barreling over some brush along some railroad tracks, and came into my view unexpectedly. I was in an awkward position amd surprised, but I swung on it and machine-gunned 4 rounds. I got the pheasant. Usually, after the first 2, you're wasting ammo. But not always.
That said, the Auto 5 is an awkward gun that gives "cover the bird with the barrel" a whole new meaning. But for a very long time, they were the only thing that worked worth a damn. When Remington finally designed a gas-operated gun that worked (the 1100), that was the beginning of the end of the Auto 5, in part because the 1100 points more like an O/U or double, and the shooter's view of the target is like an O/U. Every gun since then has been designed with a similar receiver shape, even when there's a temptation to put a big tube in the shooter's eye, as with a Benelli short-recoil gun. The Benellis have a sloped receiver rear to hide the tube.