A significant proportion of men will use custody issues as a way out of paying child support or paying the amount of child support they should. They threaten to claim custody when they don't really want it, they tie the woman up in endless court proceedings (often they can afford this and she can't because of the impact of child care on earnings), they pay erratically and just enough below what they owe to make going back to court not worth it. They use the courts prevent the woman from moving to a better economic or social support system. And that's before the threats of violence to the woman and/or children - I'll beat you up/kill you if you name me as the father or claim child support.
I'm not even including the tactics of moving to another state, leaving the country, not leaving an address for service, etc. etc. And even that's supposing that the woman has contact details for the father (rapists don't always leave these) or wasn't a victim of incest or paedophilia, when naming the father puts her at great personal risk.
Yeah, yeah, not all men, you wouldn't do any of that. We've heard it all. But too many men, too much of the time.