Remember though, it isn't usually the kids who are the targets of the angry adults. They're usually the collateral damage...or the weapons at hand. I do know people who are willing to support & interact with kids not their own, even post divorce. Or even post dating. But they are the exception.
In my personal and professional life, I've gotten to see a few divorces up close & personal- not any of them mine, FWIW- and when kids are involved, it is USUALLY a messy business. Although there is a popular idea that one bad parent uses the kids as pawns against the other, innocent parent, it is far more often the case that both sides play games. It's just a matter of degrees. (Ther are exceptions, of course.)
So, while I'm not actually a divorce attorney, what I've seen has driven me to work on some solutions to minimize certain aspects of game playing in those situations.
Of course, the is always the law of unintended consequences. Some of my solutions are pretty good for ending certain kinds of gamesmanship in child support issues, like withholding payments or misdirecting child support to, saaaay, new earrings for mom. But they're a bit slower to be responsive to issues like we're discussing now, namely, what if the kid requiring the support isn't your kid.
My understanding is that "no fault" is used when the two want a simple and largely equal/equitable split. But you are painting the picture of a guy who is really trying to get the best for himself out of this. On what planet is walking into the divorce hearings with that paternity card in hand not going to improve his negotiating position?
In the few no-fault divorces I've handled, no children were involved. Division of assets and liabilities was not equal, however. The couples usually had good reasons for the distributions they chose. But key, they made most of these decisions before engaging a lawyer to draft pleadings & petitions, etc. at hundreds of dollars an hour.
When kids are involved, though, those child support and custody negotiations are usually contentious, lengthy and expensive. And all of the other issues get examined in that context.
By leaving those issues aside until a divorce decree has been made, a father questioning paternity in a SAPCR keeps that issue from coloring those pre-divorce negotiations. He saves time and money.
It is effective enough that some divorce attorneys negotiate to include provisions in the decree that pushes the date of any potential SAPCR back a few years...at which point, you can wind back at square 1, negotiating everything over again.