I can't understand what is the problem with p-values. You should never talk about the results, the essential point is that the method provides valid conclusions 95% and invalid conclusions 5% of the time. If your conclusion is wrong (you are in the 5% part) that is not a paradox. Also you should not try to prove things since your conclusion can be wrong, you should be glad to have a procedure that gives you useful information but is not infallible. By being humble you solve the problem.