Frequentist approach gives you something solid and independent of assumptions. Probability of observing this particular dataset accidentally if there was no change between two contexts.

On the positive side, the frequentist approach doesn’t need assumptions about the pre-data probability of the thing of interest.

On the negative side, the frequentist approach doesn’t produce a post-data probability for the thing of interest either.

It provides the probability of something else - as you mention - which can also be interesting but it’s not what people really would like to know (as the generalized misinterpretation of the meaning of frequentist results makes clear).