There's a strong bias at work there though - single fathers have to be very engaged and well resourced to prevent the system taking their kids away. Whilst many single mothers just drift in said system. Only if the legal system were less biased could we hope to determine if there is a gender differ...

it is unclear. Yes in terms of correlation you are correct, but it may as well be a proxy for shitty women, also being the reason they are divorced in the first place. Widows (or her children rather) don’t show this pattern for example.