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Dun Dun Dun!!!!
The young men of the community consider a new mother who has had a baby this way to be a more desirable partner than a childless/virgin/barren woman. I believe their social conditioning overrules any atavistic disinclination to take care of offspring that are not biologically their own.
So, before I answer this question, I am both a Male and a Father - which gives me some authority to speak on this topic.
I am also a firm believer that many of our societal norms have a very real biological basis. You may disagree, but that is the position from which this frame challenge comes. Even with very strict Religious and Moral codes to offset the worst of these Biological impulses, Biology still ends up winning.
The average man's primary drive is to have sex with as many child-bearing age women as they possibly can. You might say that this proves your theory. It is my contention it does not. Children, especially small children take up a lot of time, effort and Energy. That means they are not going to have that excess time, effort and energy for the act of procreation. In short, exhausted mothers who have been up at all hours of the night breast feeding and looking after their children don't have the energy to have Sex (and we are presuming everything is consensual here).
Flowing from the statement about time effort and energy, this is true for the father also (not so much in the infancy years, but definitely later on) - Without wanting to sound like I'm crapping on step-dads (cause I'm not) - the 'reward' of parenting is seeing your kids grow and mature, in seeing the little bits of you be refined and made into a capable young adult. When raising a child that is not biologically yours, that reward is gone.
Does that mean people don't Step-Parent, of course not! but even the best Step Parents struggle for the very simple fact that the child is not 'theirs'.
Men tend to get possessive over Women of Breeding age. If you look at many of the religious cults that practice polygamy, you'll see multiple instances where all the high-status men (the Elders/Leaders) hoard all the young fertile women to themselves. And Men have the physical ability to enforce their will by being bigger, stronger, more disagreeable and more willing to use violence.
So - those are the top 3 issues that I see, and I'm not sure how you can overcome them specifically.
However - here are some things to help with your society:
1: If you want men (particularly young men) to do something stupid and against their nature, say it's Manly and make it so doing that thing makes them high-status to women. For example - fighting in a war and dying horribly is (in strict evolutionary terms) very stupid and dying is against most peoples nature. But put a Man in Uniform, with a full rack of medals on his chest and have people call him a Hero - now he's an elligable mate for Women, so his risk is rewarded.
2: Religion has had a long history of enforcing moral codes that would seem to be in opposition to our fundamental nature. I said above that Biology always wins, and I believe that - however just because countries that enforced Monogamy have long, juicy and sordid histories of infidelity, doesn't mean that on the whole the rules didn't have some effect. An enforce Religious aspect could help mitigate the biological impulse - or at least, get you into that wonderful area of story writing where you have tension between two competing ideas.
3: Women set the rules of the Game, Men play to win. Women are the gatekeepers to Sex, Men are the gatekeepers to Relationships. The latter part of that you might have trouble with - because Men will have sex with just about anyone, but getting them to stick around is the hard part - but if you take those presuppositions into account, you might find a series of systems where the Men are incentivized by the Women to do this.