The limiting factor of the speed of a shotgun (pump vs.SA) is the speed of the shooter picking up the target.
Work on the coordination of working the pump, and the only limiting factor will be how fast your eyes/brain can pick up the target.
The "slower" speed of a pump shotgun will not come in to effect until you reach the level of the exhibition shooters like Tom Knapp, Tim Bradley, and Patrick Flannigan.
Comparing two different shooters is like apples and oranges. When my buddies and I shoot (informal) skeet; I am every bit as fast, or faster, with my 870 compared to them with their SAs. But the difference is I have been shooting flying game for 25 of my 34 years. They just recently got started maybe 10 years ago (they're upper 40s to mid 50s). Can I shoot my 870 as fast as I can shoot my SAs? Yes. (And faster than my 48AL.) Can I shoot my 870 faster than others with SA. I'd give it a 50-50 chance. Can I shoot my SAs faster than those exhibition shooters shoot a pump? Not a chance in Hades.
If you want to improve your speed with a pump, shoot nothing but a pump for several years. Practice on "informal" clays, where it's one at a time and concentrate on accuracy. Shoot a clay, shoot the largest piece of that clay, then shoot the next largest piece... until it hits the ground. This will do several things, all at the same time. It will help with accuracy (shooting decreasing sizes of targets). It will help with your swing (forcing you to stay on target after you shoot). It will help you to keep your head down. Through many rounds, the speed of working the action will come.
Speed comes from limiting unnecessary movement. IE- Picking your head up off the stock. Stopping the swing of the firearm. And dropping the buttstock from your shoulder. Those are the three biggest mistakes I see with shotgun shooting. And they effect your shooting ability regardless of what type of shotgun you shoot.
Wyman