Academia and privacy
The answer to this question is extremely simple and clear-cut. No, you are not being unreasonable. You don’t want to appear on YouTube, so don’t - this is not a job requirement for a professor (or for a graduate student) anywhere. You don’t owe anyone any explanations. Do not let yourself be guilt-tripped or manipulated into giving up more of your privacy than you feel comfortable doing. And do not get drawn into a debate about the nuances of precisely which careers require agreeing to be a public figure and which don’t - this just opens the door for other people to try to manipulate you with irrelevant sophistry.

Just say no.