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You've watched people that are exactly the same skill level and the pump was faster, OK
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I have watched very skilled shooters using pump shotguns and semi-auto shotguns engage multiple targets spaced apart at realistic distances, and I have watched pump shooters out-perform semi-auto shooters on the same course of fire.
As an example:
An array of 5 reduced size shilouette targets, each spaced 1.5 yards apart. Drill is to engage far left target, then far right, and then the inside left target, then the inside right target, and then perform an emergency reload and shoot the 5th. Because of the distance you have to move the gun to perform this drill, there was ample time to work the pump when transitioning from target to target.
Even when able to engage that array of targets freestyle (all in a row) the pump users were not at any significant disadvantage over the semi-auto shooters in terms of their total times.
The semi-auto shooters could naturally empty their guns on a single target more rapidly than pump shooters...but throw in any shooting task that requires transitions between targets with space between them, or that involves making a judgment call before shooting, and the speed advantage of the semi-auto vanishes.