You are not wrong. Me and my son's mother separated when he was 11 / 12. (over 30 years ago) We came to an amicable financial arrangement; we were both happy with it and it was always paid. After a couple of years she tipped me the wink that she had to provide my name and address to the CSA. I was fuming as like many others I knew plenty of absent fathers who didn't pay a penny.And introduce rent control to cap what greedy landlords can charge. At the moment UC is simply being used
to fund landlords' property purchases and is a movement of vast wealth from the state to private individuals.
If you are salaried and caught up with by the CSA (who tend to go for the "low hanging fruit") then yes, it is. If you're a bit shifty and self-emplyed then its easier to avoid them. A friend of mine has two daughters who have children with absent fathers and neither dad is paying.
Couple of weeks later this huge package lands on my doorstep from the CSA. I wrote "not known at this address" and stuck it back in the post. Never heard another word from that day to this.
In respect of the specific case being discussed, I would suggest she gets no benefit payments at all until she has been to the CSA and through the Courts to claim maintenance from the father(s). Only if the father(s) are resident outside the UK (and have no assets in this country) should any benefit claim be considered.