Let me preface this response by saying that I watched the video without sound so he may be discussing what I am about to say. Trapping seems to be much more effective if the defender constantly shifts or shuffles forward into the opponent, as opposed to staying in the same area. It's almost like collapsing the attack. Essentially, after the initial block or trap, a straightblast would replace the trapping because the attacker's punches would be snuffed out by the constant counter attacks. Like Danny T said, "Trapping is a byproduct of striking." Again, that may be what the video was saying...without sound I know that I am missing that entire purpose of the video. I was just going by the visual. If my response has no connection to what is being shown in the video, my apologies.
Every punch should also be followed by a pull. When you punch and your opponent blocks, you can then pull his blocking arm and obtain your trapping.
When you use downward parry, your opponent can borrow your force and hay-maker to your head. This is why trapping is not good enough. It's better to grab and hold on. This way you can restrict your opponent's arm mobility.
When a fight kicks off, usually the other person has his or her hands up or in some position (whether it's near the face or more out in front). In the video, Adam initiates by trapping/blocking/knocking hands out of the way and moves forward taking his structure. Fighting someone that is punching fast and retracting punches would be very difficult I think to pull off.