And here we go again. Why is that not clapping is more legitimate than not bowing? Is that that white people religion is more legitimate than other people's?
Alejandro, it's really a matter of whether there's a Shinto shrine in the room. The clapping is done toward the shrine, where there would be a mirror and other Shinto implements. And in serious Shinto, the person who maintains the shrine would maintain it for a kami--a Shinto deity that would bless, protect and inspire the place and all the people in it. That's why it's called a "kamiza," meaning "seat of the god."
For a person who had put a kamiza in his dojo, it would be unthinkable to practice in front of the shrine and not invite the kami to bless and protect the practice and practitioners. The purpose of the clapping is to call the kami and invite him/her/it to attend the practice.
Of course, if there is no kamiza, there is no reason to clap. And many Japanese dojos don't have kamiza. In fact, Shinto is not appreciated to the same degree by all Japanese. Most of them will go to a big shrine at New Year or the birth of a child, but at its most extreme, it gets into hard right politics in which militarists and yakuza are eager participants. So it has a lot of political implications besides the religious ones.
Mochizuki Sensei was a way-back man and he had a kamiza in the dojo, containing a driftwood "dragon's head" in memory of Moirhei Ueshiba. As uchi-deshi, we sometimes changed out the green branches and dusted the kamiza. Our classes always began as Saito Sensei's class in the earlier-linked video clip, bowing toward the kamiza and clapping. Mochizuki Sensei also always led us in recitation of a Meiji essay called "Seikun". It was sort of political, but mostly centered on creating and maintaining a regular and harmonious personal and family life.
I always participated fully in the rituals and recited the Seikun, but I always prayed to the God of Israel and gave thanks in the name of Jesus. And I would love to sit again on that tatami and hear Mochizuki Sensei reciting Seikun.
Best to all.