Single mothers are, in many cases, that way because of making bad life choices. When you're poor and in a bad relationship (or in some cases not even in a relationship), kids aren't such a good idea (or at least that's how it seems to me). No one forces them to have kids (unless they've been raped, in which case I'm more sympathetic to them; how often is that the case?)
I'd be very interested to see a citation for this. You seem to be very knowledgeable on the topic, could you provide one?
Could you give some good examples of why a woman would have kids then wind up on welfare? IMHO the only 2 reasonable examples are raped women and widows.
Maybe people being stuck in shitty situations is a consequence of things beyond their control more often than your willingness to apportion blame is willing to admit.
One possible alternative is to deny them welfare, thus endangering the health of the children they care for, thus forcing the state to take the children into custody.
Financially that puts a burden on the state that now has to employ someone to take care of the children, whereas it was getting that for 'free' via the mother.
Also, being a better person should be a moral judgment not a judgement on their financial state.