Hi MrsPete. I don't think the system works like you think it does.
No, don't think about it from your cousin's situation: That's not really the same -- that's about establishing parentage from the start. Yes, your cousin was wise to do a DNA check before allowing his name to be placed on a birth certificate from a one-night stand. Here's a similar situation: My cousin (an idiot) was fooling around with a woman who was separated but not divorced, and she got pregnant. She was still married to the other guy when the child was born, and
according to state law that guy's name had to go on the birth certificate -- even though all three admitted, "Yeah, J is the father." I actually work with a guy who -- for the same reason -- is the legal father of two girls, even though he is only the biological father of the older one. Ridiculous, I know.
The situation proposed is that the unmarried couple has lived this way for a while -- long enough to have a couple kids -- then things to belly up. If the woman has the money and the wherewithall to take Sir Frugal to court and demand a DNA test, yes, she can have it -- but, remember, this hypothetical woman
has no money of her own. Let's say she does have money to take him to court; maybe her parents or siblings are helping her. It's a tough argument to make:
"Yeah, this guy, who has been living right here in town all this time and working in the same job for years, yeah, I just figured out he's the father of my three children." The people who work for Social Services aren't idiots; they'll know she's known the father all along. Anyway, she files the paperwork ... and she's assigned a court date two months from now. Maybe she has government housing, but
now she really is living on food stamps. And if he's trying to be a jerk, he goes into court and asks for an extension (which will be granted once), so now she's dragging on for months before she can go to court. Yes, the court will order him to submit to a DNA test (assuming he hasn't left the state or disappeared), and then after that's processed, they can go back to court to determine child support. By the time this actually happens, a year'll be gone. After all that, hopefully he will pay his child support ... but that's not a certainty. If he doesn't, she has options, but -- again -- those options take time, and she can't do anything 'til she's proven that he has a pattern of not paying. Furthermore, if Sir Frugal has retired early and has no real income, she's stuck; as I said before, her fake poverty could become real poverty in a hurry.
Maybe you're thinking that the state will help her -- nope. If she's been claiming for years that she doesn't know the children's father, and she's been supporting herself on X amount from the state ... they're not going to suddenly agree to help prove that Sir Frugal's the dad -- she's already on the dole, and they don't want to go back and re-categorize her. Plus she's been telling them for years that she's supporting herself on the money they've been giving her; going to the state for help means 'fessing up that she's known the dad all along and potentially opening herself to legal prosecution.
Also, consider the kids' participation in this situation. They're going to be dragged into a lab to have their cheeks swabbed to prove who their dad is. No mom wants to put her kid through that; it's different when you're talking about a baby who won't remember.
No, this we're-going-to-live-high-on-the-hog-while-on-the-government's-dime plan isn't a wise situation for a woman (or her children). Too much risk.