What you propose is so drastic that it would never pass muster in today’s society. For one thing, both men and women are irresponsible so many women would just abandon the child or get abortions. The abondened child would be thrust upon her parents (which happens in large part today) or put in the government system of foster children/etc...
Basically that option is not that good either. It’s a bad siuation all around. I would propose that men get more rights in regards to being the primary parent. Why is it assumed that the mother will be the best parent by the courts?