Custodial parents who choose to cut non-custodial parents out of their children's lives will also be made to understand that they are making a choice to live without child support, and by the same token, non-custodial parents who make the choice not to be part of their children's lives will be making the choice to pay child support. The key word is choice. In order to be on the hook for child support, the non-custodial parent has to have made a choice not to be a part of the children's lives. Custodial parents who purposely cut children off from their other parent should have to bear the burden of caring for their children alone, because they have made that choice. Both parents need to be responsible not just for caring for the children, but also for their own choices in dealing with the children and the other parent.
An unmarried father has one month from the time that his child has had a DNA test to confirm that he is the father to relinquish custody, at which time he will not be forced to pay child support, but will also not have legal access to the child, and the child may be adopted by a man that the mother marries if he should so desire.
No parent who has had their rights involuntarily terminated will be forced to pay child support. They do not have access to their children, and therefore, the children, by law, are no longer theirs. As such, there is also no financial obligation.
Child support orders may not be made retroactive. A new order will only be paid from that point on, not from some other date. By the same token, if a parent has receipts for things they have bought for the children, that will count as proof that they have been paying child support.
Non-custodial parents may request at any given time to see receipts that state what is being spent on the child(ren). This can include rent over and above what the parent would need for a single person, utility bills, food for the children, clothing, etc. If the amount spent on the children is not at least twice what is being received in child support, a modification may be made to reduce the amount paid in child support.
Arrangements for child care will always begin with asking if the non-custodial parent can provide it as long as they are in the area, and the non-custodial parent will not be responsible for any child care expenses incurred at a time when they are available and willing to take on the care of the children. This will also count against the non-custodial parent's child support obligation at 1/2 the going rate of child care in the local area.
In cases where there is more than one child, child support orders will automatically reduce to either the state minimum or the percentage for the number of children left, as each child reaches the age of majority or graduates high school (whichever is last).
Child support will automatically be modified upon the loss of a job to the statutory minimum until another job can be found, and then will resume at the percentage set by the state of the new wages for the number of children involved. Providing child care, clothing, food, school supplies and other necessities should count toward child support obligations as each of these reduces the burdens on the custodial parent. A loving parent who is out of work, can easily provide childcare for their children, and that needs to count toward child support paid at 1/2 the average rate for their local area. Imputed income will never be an option for either parent.
Jail should never be an option for parents who are unable to pay child support, even as a last resort. Losing driver's licenses, passports, and professional licenses should also not be an option, as each of these losses reduces a person's employment prospects, and is therefore counterproductive.

I believe that the above remedies will greatly reduce the number of so-called deadbeat parents, and allow parents to just be the best parents possible. No parent is ever going to be perfect, but this will help even single parents and parents who are divorced to do the best they can. Child support and custody should not be the war that it currently is.
This would actually be in the best interest of the children and the parents, because it would allow the children the opportunity to love both parents, and it would make parents who try to cut the other parent out of the children's lives stop and think before doing so, as they would lose the opportunity for child support by doing so. It would make it to their advantage to show more kindness to the other parent.