For the shift that the author desires means society and women when making that decision to pull the relationship trigger needs to set the boundaries that relationship equality and parental equality are the standard.
That means entertainment and sex appeal in society needs to show that the stay home at dad and the "Dad" mentality for men is considered the hot and sexy standard.
Right now society doesn't value this from men, and men have opted out, and the shaming that is done currently pushes men away from the discussion and amps up the "ugh" factor and shuts them down.
IT is a simplistic answer, but men do have to adult up, but if men walk away and shutdown, that can be an acceptable response as they are stating they aren't ready.
In my world view, society may need to create a professional parent status which is funded by society to cover this and make it appreciated the costs and value of parenting.
That would be my suggestion to the madhouse of Americana. You may get men who don't want to work as an enslaved male to society, want to take up parenting as suggested if that option existed.