I've spent my career thinking about these same issues. And certainly Fisher was, well, abrasive to put it mildly. I've read much of his early work and I think the intended context has mostly been lost. Many modern statisticians ( of the "frequentist" persuasion ) use a strange and awkward combination of fisherian and neyman-pearson. We talk about "p-values" but then interpret them as hypothesis tests with long-term error probabilities ( Fisher disavowed this interpretation of p-values, and insisted they did not have a long run probabilistic interpretation but were a measure only of evidence against the null in the particular experiment. ) I think Fisher gets a bad rap for a lot of later bastardization of his work. ( I'm sure it did not help that he was not a likable person, or his valid but misguided and testimony about tobacco. )

Still, I'm sympathetic to your position and can understand how you'd come to that way of thinking. I'm not convinced even a diehard Bayesian would completely disagree with Fisher's more restricted and stringent interpretation of p-values. Or at least they would see it as a big step up from the more common usage you are referencing.