Could this not be a symptom of the hard fact that women do get custody more often and the general perception that it's difficult for men to get custody? If men are most likely to try in cases the woman is so pathetically incompetent, abusive, or criminal it's difficult to imagine not revoking their custody, even sometimes to the point of placing children with fosters or relatives if the father weren't interested in custody, then certainly it would be true that men get custody "when the bother" trying. However it would also be possible that plenty of men do not try because they see no hope. It's definitely possible and I personally believe it's likely that the majority of men who would like custody end up not trying.

I've had this impression many, many times when fathers talk about trying to get custody, and in my own life.

My own brother in law has gone to jail with obvious defensive wounds and not a mark on his visibly intoxicated ex-wife...he still can't get custody of his child on top of the criminal system willfully ignoring the evidence of their fights. My step father couldn't even manage joint custody when my mother was a diagnosed borderline and a severe alcoholic - usually neglectful, extremely emotionally abusive and sometimes physically abusive. This kind of extreme privilege for women in family law happens day in and day out across the whole fucking country and we still have people like you arguing, effectively, that "male privilege" dominates even in family law and domestic violence cases. It's absolute bullshit.

What you're saying doesn't prove a male privilege, or that women do not have privilege, in family law. I maintain it's only an ignorant or willfull misunderstanding of which subset of divorcing men do seek custody and why.