You're mistaken. Paying reviewers would only cause a conflict of interest if it biases reviewers in a particular direction. Which direction would that be?

Paying the reviewer doesn't incentivise them to accept or reject only to actually do the review. As it stands, in many fields reviewers are far from experts in the area they're reviewing but only in the same general discipline. Papers sail through review without proper statistical oversight, because what statistics expert is going to spend their time reviewing psych or bio articles when that's not their discipline?

Well if we paid people to do what amounts to some of the most important work in science, maybe we'd get actual experts to do it!