A woman in England went to court for the right to keep a one night stand from being notified that she gave birth to a child from the episode. Now maybe I am not a person to understand the nuances here. I have never had a one night stand. I can't imagine what it would be like to do so and find out that the consequences were going to be another human being.
I can, however, categorically state that I would not want to have to put up with a long relationship with someone who meant so little that once was enough. I would not want to hand a helpless person to someone who could walk away that easily. I would, like the woman in the article, want to make sure that someone so casual had no say in what I did.
Father's rights? No. here we are talking sperm donor rights, which is a different thing altogether. Just because the donation was made in person is no reason to give it more weight than a donation made in a lab.