Is the pp-value a good measure of evidence? Asymptotic consistency criteria

What are the criteria that a measure of statistical evidence should satisfy? It is argued that a measure of evidence should be consistent. Consistency is an asymptotic criterion: the probability that, if a measure of evidence in data strongly testifies against a hypothesis HH, then HH is indeed not true, should go to 1 as more and more data appear. The pp-value is not consistent, while the ratio of likelihoods is. The same holds also with respect to the unconditional consistency criterion.