This is not what they said. They say "falling more causes more stress on the gear" which is correct. The caveat here is that falling is meaning free falling due to additional slack in the system. This is wholly different from a soft catch which does not add slack but distributes the force of the catch over a longer period of time, therefore breaking while "falling more".
Please note one thing on the video:
Unnecessary slack in general is not regarded as something that is helpful for lowering the impact. More slack creates more conversion of potential energy before the rope catches which you then have to shed again through friction. Therefore it is always better to avoid adding energy to the system.
Why do people then think a catch gets softer with more slack? Because the higher impact at the begin helps to overcome the inertia of the belayer. This is especially important when you belay with a locking device and can only give a soft catch by moving with your whole body. With a dynamic device this is different as feeding more rope through the device will give a dynamic catch.