In a perfect world, sure, parents would do what is right. But here we are.
I just don't think more red tape is the answer, especially on something that furthers the burden on the taxpayer and not the person trying to stick up their child.
Some solutions I would have, firstly stop treating sex as transactional and polygamy as normal and raise your children not to participate in such things.
If they do participate in such things and have to face the results of their actions they need to be prepared to take it on as a single parent and know full well that could happen before hand. (Not saying to live your kid high and dry but they should expect such if the situation arises, obviously you shouldn't have a starving infant and mother/father on the street)
Hold the parent that leaves the relationship as responsible, a lot of my friends parents who were "supported" by child support skated by with making near zero fiscal contribution and also they were detached from raising their child and looked at the CS as their only obligation.
I'm not sure how else to hold them responsible other than community backlash, which would essentially have the same effect as a govt mandate with no taxpayer burden.
The current system fails, and a few people generationally live off child support and disability/wic that I personally know. I live in the poorest state in the us, it breaks my heart to see it being perpetuated even further, eventually someone has to pull the band-aid off we can't keep operating at such a massive deficit for much longer. Eventually the free money runs outs.
We live in a world where people get sucked in to the debt train and can never escape. People live beyond their means, have to have the newest $2000 phone, the $100 jeans $500 shoes $50000 car etc to look rich but it's a facade. Many of these same people live on CS and state benefits including medical cards, wic, etc..
I don't have all the answers on it but that's my .02¢ I'm sure it's worth about .01¢>