I think it's only 'normal' for those folks who are on 'The Baby Story' (a TV show in the US). My birthing coach (back in the dark ages) strongly suggested that we ask for no visitors until baby and I had been home for a few days. Our parents, etc., honored that.

My daughters (neither of whom are any where near providing grandchildren yet :( ) have both told me they have no intention of having an 'audience' in the delivery room. They don't intend to expose themselves to anyone but the essential medical personnel and their husbands. Seems quite reasonable to me!