This might not be an interesting thread. But what works for me, in terms of explaining the link between welfare and single mother rates, is the idea that welfare makes it possible to be a single mother.
The idea that people were becoming single mothers in order to get benefits never took of on me, because the child is often more expensive than the benefits.
What makes sense is that the welfare state makes women able to become single mothers, and they do so because they want to have children but don't want to be married or are unable to get someone to stay with them.