A pump action shotgun can be pretty darn fast. I shot a round of wobble trap a few months ago with my Rem 870...everyone else on the line (including a few members of a skeet/trap team) had over/unders (you cannot get faster follow up shots than with an O/U or a SxS). Anyway - I was able to hit one of the clays twice on the fly. I also had the high score - so go figure.
I find that the act of running the slide forward helps bring the muzzle back down on target.
Now, Mas Ayoob also recommends the semi-auto shotgun - in his case to eliminate the element of human error (no risk of short stroking the action).
I will say that, other than extraction issues with cheap Winchester X target ammo, my Rem 870 has had no issues. I cannot say the same of the (expensive) semi autos that my usual shooting buddies use. FWIW.