I'll add my two cents because I probably do things a little differently.
I think they don't get depressed the way an isolated bully might, precisely because they are broody -- they are almost in a trance during this time, and sit still for hours.
But I usually leave them with the flock, because I have the time to check several times a day for added eggs, etc., and because I want her to raise the chicks with the flock, to avoid integration issues, and because I want a flock that can do things the "normal" way -- or, I want them as independent as possible. When I've put the broody in a separate pen, I've opened it up a day or two after the chicks hatched. I do nothing about broodies or new chicks, really, other than make sure there is food and water they can reach. I feed the flock one food they can all eat, usually grower or flock raiser, then provide grit and oyster shell separately, making sure there is some fine stuff in the grit container for the chicks if they want it. Even in below freezing temps the chicks do fine using mama for warmth. I'm convinced they are healthier this way. You'd be surprised how much time even a very young chick will spend out pecking and scratching before getting under mama to get warm. It makes me wonder how they survive those 95 degree brooders. And they never get a pasty butt.