As far as I know, the Higgs boson was already part of the standard model since 1967, so the usual structure of hypothesis testing distinguishing H0 vs H1 should have been foggy. In any case, besides rejecting the model without Higgs, the new particle had exactly the features predicted by the standard model. So it was not only to reject H0, but to find something really nice by induction.
What would have happened if there had not been a previous theoretical framework? Maybe the experiment would not have been done. In any case, theoretical physicists would be right now thinking about its deepest meaning, trying to arrive to the Higgs underlying theory, but at the same time, the engineers would still be checking that the wires were properly laid.
As far as medicine is concerned, everything seems even more ambiguous, as you say, because the evidence is usually much weaker, not being able to perform such rigorous statistical experiments on many occasions. In addition, there are many more incentives to make disparate interpretations, since physicians must make urgent decisions about their patients at all times.
This is why I have the feeling that a p-value is just an attempt to build a useful cognitive aid, to guide us while we try to learn difficult things.