I believe there is a simple, yet profound explanation for both of these facts, such that the criticism is unwarranted. The role of skepticism begins with the following basic principle, let us call it the principle of assessment. When a body of science comes to have public policy implications it must undergo a higher level of scrutiny. The rationality of this principle is obvious. In pure science a big mistake is seldom harmful, but in public policy it can be disastrous.
Pure science encourages speculation, and following promising but untried lines of exploration.
Public policy, because it mobilizes vast forces, necessarily operates under a far stricter standard of certitude. The role of the skeptic is to implement the principle of assessment. One does this by testing the claims, probing the foundations, cataloging the uncertainties, seeking out disagreement, etc. In short the skeptic is an investigator, not of climate but of climate science. The scrutiny is not part of the science.