Hopefully someone who had a child at that time can answer. It's a little before my time, but ...
I recall an 'All in the Family' TV episode, circa 1972 or so, where Gloria, Archie's daughter, gives birth and her husband is present.
It was about this time that women were taking childbirth classes with their husbands. (Or boyfriend, sister, or someone close to them.) I was teaching in 1974 and this was all the talk whenever a teacher was pregnant - did you want your husband with you? Why or why not? Oh, I'd never do that! Or yes, I want him with me, why not?
So if not 1968, it was shortly thereafter that this (dad in the delivery room) became more common in the US. Prior to that men were often found (on TV, movies) pacing in the waiting room, waiting for the news: is it a boy, a girl? Is it healthy, that's all I want to know.
I did have an aunt who had a baby in 1967 and she had it (her) the way I mentioned above, husband not with her, but outside in a waiting room. It was her fifth child though, and possibly she just continued having babies the way she had the previous four.
Sorry for so much digression and hardly a positive answer.