It really needs no explanation to prove that recoil only happens after the bullet leaves the muzzle, just watch the end of the video where he fires a revolver. You will clearly see that the muzzle doesn't move until after the bullet leaves the muzzle.
The mistake most laypeople make is that they don't understand the basic application of Newton's Laws. In order to apply Newton's Third Law ("When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.") you have to define the system that you are applying the law to. In other words, when you apply the third law to a gun and it's recoil you must only look at the parts that make up the system. In the case of the movement of a gun the system is the gun and the cartridge, we usually exclude the shooter because that just complicates what is happening with the gun and the bullet (the bullet being part of the cartridge)
With the system being defined as the gun and the cartridge we can see some pretty obvious things right away;
1. The bullet is far lighter than the gun.
2. Force of the hot EXPANDING gasses must be included in the system.
I emphasized the word expanding because the act of gasses expanding creates force if it's opposed by an object.
At any rate, as the bullet moves down the bore all of the forces in the defined system are contained inside the gun. On the one side of the equation you have the expanding gasses and the bullet movement that apply forces in one direction and then you have the counter forces of the gun's mass on the other side of the equation. The two sides of the equation equal out and the center of mass of the gun doesn't move. Every force that moves in one direction inside the gun is opposed (as per Newton's Third Law) by an equal and opposite force, that means that it's impossible for the center of mass of the gun to be moved in any direction by the forces inside the gun because all of the forces inside the system (the gun) are opposite and equal, there is no resultant force any direction.
But when the bullet leaves the muzzle you change where the forces are applied, they are no longer being applied inside the system, now you have force being applied from outside the gun and the gun can be moved by this outside force.