[spzb]

The buried lede in that link is that mothers who don't have custody of their children are more likely to remain in close emotional contact with their children than fathers are when in the same position. So children living with dad still benefit from having both parents involved in their upbringing. Which undermines OP's assertion that this child would be better off without their mother around.

[giantg2]

Yes, involvement from both parents seems to be the major factor regardless of sex. There is likely additional research needed on why fathers disengage more when the mother has primary custody. With a majority of single parent households being headed by mothers, it seems another area ripe for research is how unlikely it is that the majority of fathers are disengaged to create such a large effect on the whole single mother cohort. Likewise, with the way custody tends to be grated in court, you would expect single father households to have a higher percentage of unengaged mothers due if it was determined that the mothers were the lesser choice for child welfare. I would guess looking at outcomes where one parent died would mostly control for that support mechanism.