I don't believe many people do use the CSA out of vengeance, or spite, or whatever you want to call it. Some, yes I'm sure, but I think they are in a minority.
There are 2.5 million single parent families. The CSA is responsible for making 861,700 of them pay. That's 34.4% of NRPs in the whole country that the CSA is gathering payment from. More than half of RPs don't go to the CSA at all.
Of the 861,700 that are paying nearly half (47.3%) are only paying £5 per week regardless of how many children they have.
And I don't think going to the CSA when the NRP is already making payments is a sign of greed either. If the NRP's offer is less than the CSAs calculation then the NRP is the one being 'greedy' (and selfish) IMO. The CSA recognises that it is not always possible to get people to pay a fair proportion of what it really costs to raise a child, because people on low or no incomes can never pay their fair share without making themselves destitute. So what the CSA does is determine what is a fair proportion out of the NRPs income. If an informal offer is well below what the CSA suggests, it is not greedy for the RP to go to the CSA if discussion with the NRP gets nowhere. In fact, you could argue that it is the child's right for the RP to do that and maximise financial security for the child.
In an ideal world, and obviously barring cases of abuse etc, we'd have a situation where contact was unlimited and NRPs paid without fail. Sadly there are bad NRPs and bad RPs, but if you're looking at the currently available evidence, the number of NRPs not paying anything is by far the much bigger problem. That doesn't mean the other problems should be ignored, but if we're going to put children at the centre of this, and make it all about them, we need to prioritise the problems.