Yes the children need their Father but the point is that it has nothing to do with whether the woman goes to work, that is two separate issues.
If they go to work the father still has to pay support but at least the woman is than taking responsibility to care for herself and her family.
The separate issue is whether there is a public interest in compelling payments from fathers who have not abandoned their children to the welfare roles. This is the means test Phyllis is describing here. The program legitimately compels fathers who do abandon families to the welfare roles to contribute or reimburse the government for public assistance payments they are obligated to provide when they had the means to do so.
Assuming the people gave the power of parens patriae to their state, and it could be argued they did not, the power is at least limited to cases that demonstrate a need to protect a class who are abused or neglected. Without proof of such a need, the state has no power to interfere with our right to be left alone. The issue is this program ignores any limit to the excersize of government power to interfere with private rights.