The Chareidi Modern Orthodox, not sure what this has to do with korbanos, but…
Yes. This is the old minhag.
As far as I understand and I’m not sure how long ago this was, only the Rov used to wear his tallis over his head. The tzibur all wore hats for davening, (like bochurim do now, on top of their tfilin) and talisos just from the neck down. Gradually, from the litvishe yeshivos, it spread that all married men wear their talisos over their heads. I greatly respect people who keep the old minhag (although I don’t), and you don’t see many of them any more, at least around where I live. Firstly, because the ‘more frum’ have taken on the ‘litvishe’ way, and others have stopped wearing hats in the weekday for davening.