It should have ended as follows: "...could revolt and stop paying taxes that help fund the education of the children of people who have children." I hope that you will ignore the original and accept this modification.
The difference is that taxes are levied based on property or income by everyone.
Are you suggesting that, provided that everyone were to pay at least one penny per month into some general fund for child support, then you would have no objection to the proposal in the Original Post?
The purpose of child support is to provide for one's children and encourage responsibility.
What would prevent the "encourage responsibility" part from being given as a justification for the proposal in the Original Post? Military conscription in Switzerland -- which can cost a person his or her life -- might be justified partly on the grounds that it encourages all citizens to take responsibility for the security of Switzerland. Why can't the diversion of excess funds -- that are already required to be paid by law and would not be increased by the proposal in the Original Post -- be justified on the grounds that it encourages citizens to recognize a general responsibility for children?
Having payment go to other children would encourage other non-custodial parents to avoid their responsibilities.
Fire extinguishers could conceivably encourage people to play with fire. Would you propose removing all fire extinguishers from public school buildings? Did it occur to you that some children might have unsatisfied needs simply because some non-custodial parents have low incomes, even though those non-custodial parents may be paying everything they are legally required to pay, plus extra money that they are not legally required to pay?