I set up an experiment this afternoon using two otherwise identical Benelli M2 twenty gauge shotguns, except one with the open sight arrangement pictured above and the other with a T1 Aimpoint. The "test" was to start from a low ready, and to hit three eight inch steel at 15 yards, spaced approximately five yards apart, using buckshot and high brass birdshot. I repeated the test about a half dozen times with each different shotgun.
The times were very close, with the iron sights perhaps a smidge faster. A smidge meaning maybe .10 over an average 1.5 second run. However, I didn't miss a steel with the red dot and I had some misses with the open sights. Then, I went back to 35 yards and shot head shots on an IPSC target, and as you might expect, the T1 was an advantage.
I am going to try to repeat this test, and make my wife a guinea pig to do the test, too.