I once asked my rosh yeshiva why we are noheg to listen to music on a regular basis, even when Rav Moshe held differently. We also discussed why we sing psukim, and especially parts of shir hashirim, when this is also questionable from the gemara and rishonim.
He answered in two parts. 1) It is a mitzva to be melamed zchus on klal yisrael, and 2) music today has a different social context than music did 200 years ago. Until recently, music was part of an occasion. You had to hire live performers. Nowadays, music is background noise everywhere we go.
Based on the above, as we know that many, many leaders of klal yisrael appear in these Tisha B’av videos (and they know that these videos include some form of background music), and the music is not upbeat but rather serves to enhance the feeling of mourning and loss, there is good reason to assume that the background music is deemed acceptable.
This isn’t to say that you should run to listen to sad music. But in many shuls, they sing parts of kinnos, or between kinos. If these videos help people experience Tisha B’av in a more meaningful manner, and the leaders of their communities embrace them, the YWN coffee room may not be the right venue for a productive debate.