Royalty in itself is a bizarre concept for 2020. I think many Royals understand there are certain things that must be done "to keep their subjects happy", in exchange for the privilege of a life that wasn't earned through merit, talent or intelligence.
I don't think it's THAT much of a sacrifice to pose for a photo when leaving hospital. It's a joyful occasion and despite being intimidating on a personal level, those few minutes can leave a lasting impact on millions. I suppose the point is that many people have jobs where you need to "buck up" and get on with things despite feeling less than up for it. I think it was much harder on Kate when she had to attend Christmas service with morning sickness (George).
Of course childbirth gets turned into a question of womens rights but in the case of Royalty it's more a question of professional obligation. You can argue that people in that stratosphere of privilege do not owe their subjects anything. But this won't make them any more likeable or relatable (case in point PA). If you're leaving the hospital anyway with a baby, what difference would stopping for 20 seconds to smile for a photo make? In Anne's case she didn't even care but the public still got their shot and was satisfied to some extent.