I have no idea how a researcher is supposed to show that CP induces demand in pedophiles without running into some pretty massive fundamental issues. There are already troubles getting people to admit who they're going to vote for anonymously, so getting people to admit that they are using child pornography anonymously seems like an extremely tall task.
I've also seen studies trying to show the opposite, or trying to show that there is no causation either way, so I don't think there is anything resembling a consensus on this issue at all.
What research? The research with correlations that people who look at that are statistically more likely to do things?
Sure. If you take a sample of people with absolutely no interest and a sample with people who do have interest you would find the sample with interest would be more likely to attack someone.
Does the consumption of CP drive the desire or does the desire lead to the consumption and subsequent unwanted behavior in some people? Is punishing people for their desires and driving them underground really helpful? All to sate our sense of disgust? Is this even compatible with due process?
Ideally, you would advance technology such as to make artificial alternatives indistinguishable from reality and produce robots. Or whatever it takes. Disgusting? Sure. But, it is better than the alternative and it preserves the high standards we set for our liberal democracy.
You keep out of other people's business and they keep out of yours. No one tells you how to live your life, so long as you don't bother them.
It probably is at least somewhat. Violent video games have also been shown to increase feelings of aggression in lab settings and in some cases even foster sexist attitudes. I mentioned this to you in another comment, but Gary Young's The Gamer's Dilemma book is a very good book-length treatment on the issue of virtual violence vs "virtual pedophilia".
I've also seen studies trying to show the opposite, or trying to show that there is no causation either way, so I don't think there is anything resembling a consensus on this issue at all.