P-values are not comparable across different null hypotheses, they are a property of a particular sample given a null hypothesis. You can compare p-values only

P-values are not comparable across different null hypotheses, they are a property of a particular sample given a null hypothesis. You can compare p-values only by making very strong assumptions that the samples come from the same data-generating process. In almost all cases this is unlikely. A frequentist usually builds a more descriptive model by adding more variables (accounting for obvious factors that contribute to differences in observed frequencies) and increasing sample size to increase statistical power (reduce false negatives). This is not applicable in this case because there is only one president and the tests have low power. The test is bogus because you can't 'sample' and determine to probability of X==president effectively.