You might want to look closer at history. Many men abandoned their children. There was a word for those abandoned children: bastards. And there was plenty of widows and orphans. Funny how the idealized version of patriarchy tend to forget that many women did had to raise children on their own through all history.
On the other hand, you demonization of single mothers today, is based in no facts. Many childrens from single mothers are perfectly normal, while some children with abusive fathers are scarred for life. Sure, there are healthy marriages too. Bot heterosexual and gay. But I'm afraid that the idea that without a "father role model" a children would "not be socially integrated" is not true. I'm sorry if that hurt someone's ego, but there are plenty of studies that show that what affect children is family's income. You give single mothers resources, and suddenly, there is not problems for those children. Surprise, surprise. Google the studies.