The fundamental problem is simple. When people do statistics the answer they want is, "How likely is it that X is true?" The difficulty is that the problem is ill posed, you lack sufficient information to answer that question.

Classical statistics replaces the question with one that can be answered. Namely, "How likely would this result be if the null hypothesis were true?" This has several difficulties. The most noticeable one is that, no matter how much the professor tells them not to, people replace the question actually answered with the question that they want to answer. This mistake has been made by anyone who says, "We confirmed the null hypothesis..."

Bayesian statistics confronts the problem head on by pointing out that the conclusion you should draw depends on the prior beliefs you start with. And then they present complicated graphs that show you how your prior affects your conclusion for some reasonable family of priors. This approach avoids misrepresenting the question at the cost of presenting your answer in a complicated way.

My feeling is that as long as the general scientific public remains unconvinced that the classical hypothesis testing approach leads to wrong results, simplicity will win. (And will continue to be misunderstood.)