Now it is more a matter of my authority.
Well, yes... I’m sorry if this will come as a surprise to you, but coming across as an unreasonable, coercive boss who wants to force their students to participate in distasteful, privacy-violating activities that have zero relevance or value to their professional training, is in fact something that will greatly undermine your authority as an advisor. Effect, meet cause.
My suggestion is that, for your own good and the good of your students, you cease those authority-destroying activities immediately. Your “authority” is something that is given to you for the sole use of doing research and training and educating your students. Asking them to appear in social media videos so that you can have a “visible start” is no different than asking them to make you coffee or give you a back massage so that you can feel energetic - both of those things are indeed not in any job description of any PhD student anywhere. They are not things you should ask students to do, and they are not things you have any right to be upset at a student for not wanting or agreeing to do.
Finally, the fact that you asked the student in the interview if he would be willing to appear in YouTube videos only means that you declared your abusive intentions in advance. It does not make those intentions any less abusive.
Good luck in your tenure track career.
Edit: I said I will address some of the comments left below. Rather than try to discuss fine points of logic, which I think will be tedious, I've decided to add a more personal type of note, which I think will do a better job of getting the point across.
I have been asked many times over the years by various people to have lectures I was giving recorded and made publicly available online (on YouTube, etc). I always refused. Frankly, I find the thought of making publicly available videos of myself lecturing pretty repugnant. Why? Well, just because; I don't feel like it, and I don't owe anyone any explanations. If some silly people want to think that it's because (to quote a comment someone wrote) "some people are super self-conscious about their voice and/or their looks", fine, go ahead and think that. (Oh, and by the way, on another occasion I did agree, happily, to participate in a YouTube video that now has more than 700,000 views. Go figure... I guess I'm inconsistent! Again, I leave to others to engage in their pointless Freudian analyses of what that means about my personality or whatever.)
One of the lessons I learned from these experiences is that people who don't care about privacy have a really hard time understanding people who do. The people who asked me to make my videos available online were sometimes quite offended at my refusal, as if they could not understand why anyone might refuse such an amazing opportunity, and even (this was really bizarre) subtly pressured/guilt-tripped me, implying that I would be depriving the world of an important resource (trust me, I wasn't).
OP, I am sure you are a good person and a good researcher who means well, but know this: your student who doesn't want to make YouTube videos really, really, really means it when he says he doesn't want to make YouTube videos. You might have a hard time understanding why; well, it's not important (or any of your business really) why -- just trust that it's true. And it's quite possible that your other students who haven't been as firm in standing up to you feel exactly the same.
To summarize, you probably don't view yourself as an abusive adviser. You think the student is being unreasonable, but he isn't. Attempting to force the issue will have bad consequences for both of you, and more importantly, is simply wrong. So please let the matter go, and figure out more acceptable ways to market yourself and your group.