Divorce propels a large number of women into poverty. "75 percent of all women who apply for welfare benefits do so because of a disrupted marriage or a disrupted relationship in which they live with a male outside of marriage." I wouldn't exactly call that favoring women.
https://marripedia.org/effects.of.divorce.on.financial.stability
So, saying that men don't initiate divorce because it will cost them more doesn't really tell the whole story. That may be a factor for some richer men, but that needs to be balanced out with the fact that many women are so unhappy that they'd rather be poor than married. The Rosenfeld study bears this out - with most of the women from the study reporting that they were sick of the outdated gendered expectations that still exist in heterosexual marriage where women still do 2/3rds of the home and childcare, as well as the sense that their men weren't willing to work on the relationship. One of the conclusions of the study was, "One paradox of gender, marriage, and the life course is that young single women appear to desire marriage and commitment more than men do, yet married women appear to be less satisfied with their married experiences than men are."
Family courts tend to award primary custody to the parent who was providing most of the care for the child prior to the divorce. Quite often this was the wife, even if she worked outside the home. "One factor in determining custody is which parent has been the primary caregiver for the child. Some states actually use the term "primary caregiver"; others refer to the parent who is best able to meet the child's needs, who is most willing to accept parental responsibilities, or who has been caring for the child."
A lot of men have minimal involvement in the hands on day-to-day aspects of caring for their children, so it's not that big a change for the woman to find herself a single parent since in many ways she already was. Lots of men are great dads and family courts no longer reflexively favor women the way they did 20 years ago, but the fact still remains that in most families the person most "responsible for meeting most of the child's daily needs, such as feeding, bathing, playing, waking and putting to bed, making doctor appointments, arranging for child care, and so on" is the mother.
If you really are a lifelong believer in equality, you might want to fact-check your premises before you assert them because some of them don't really align with the facts.
