Every single last law is sexist.
We do not need to debate the theoretics of it, the statistical facts are women are better off after divorce.
Vastly more women receive custody, child-support, etc...
In *any other* aspect of our laws Affirmative Action would apply and say if the law creates such an imbalance, 95% to 5% women/men custody, then the "law of big numbers" tells us unequivocally that it is obviously and overtly sexist.
70% of all divorces are filed by women; obviously sexist.
Statistics also tell us that children raised by single fathers fair better in life than those raised by single mothers. The "tender years" doctrine is not merely sexist it also appears to be incorrect.
The arguments you are giving are anachronisms from before women's suffrage when a man could leave his wife of forty years who never worked a day in her life and /someone/ needed to take care of her and the State didn't want to pay. Or more to the point, she had to choose between a life of destitute or accept a philandering husband.
"Spousal support" has supplanted alimony and is generally limited.
Today both men and women are (suppose to be) responsible for themselves but the message every step of the way is men are too incompetent to have custody yet women are too incompetent to raise their own children without help.
The default judgment ought to be:
Joint legal custody.
Shared physical custody.
Shared responsibility for child-support; both pay 50% into a trust for the child.
Both parents are responsible for maintaining residence for their kids; no support is paid for this.
Circumstances can then alter it from there; e.g. if one parent decides to move 'too far' away they have to give up physical custody and an adjustment to support payments made (not flip from 0 to 100% or 50% to 100%)
The cost of raising the children should be the determining factor for support payments, not a cash-cow based on income. People who make little money would not "get off easy" and people who make a lot of money would no longer be taken advantage of.
If the man works and has a decent job and mother does not, physical custody should be awarded to him, even /pressed/ upon him.
Then the mother will have the time necessary to finish her studies and get herself on her feet and she should have 'right of first refusal' to spend as much time with her children as she desires.
Child-support is NOT for the child; child-support is for the mother on behalf of State.
She receives the money.
She has total freedom on how it is spent.
If he dies, she is still responsible for raising her children and if that happened then the State would pay her from welfare programs.
Child-support is, in reality, in fact, for the mother.