Yes a pump will throw off your aim as you cycle the action, depending on the experience and skill level of the person though it may be little more than what one gets from the muzzle rifle as a result of recoil.
Also, when shooting most shotguns <at least the ones I own> the bolt unlocks after firing and the slide is already to the rear or mostly, so the firing just needs to push the slide forward chambing a new shell and brining the weapon into battery, and all of this can be done in seconds while simultainiously bringing the weapon onto target.
We must remember, these are shotguns we are talking about, not precision weapons with pinpoint acuracy.
As for bolt action, yes, it is easy to keep on target when firing, the firing hand that works the trigger also works the bolt without moving your cheek weld or the forward supporting hand.
The same can be done and easier with a lever action. Again, you do not remove your cheekweld from the weapons stock and mess up your sight picture, but also, the hand does not move from the trigger to the bolt, it simply drops the level and the hand returns to the trigger as a result of working the action.