Yes, the calculation of a p-value is always done by assuming a specific model, though usually a less controversial one than what is proposed. But I wouldn't go so far as to demand smaller values.
I would prefer we stop thresholding so much altogether, and instead operate with the understanding that what has or has not been found is a suggestion with evidence for it, rather than fact, for all but the best-understood processes.
It is a difficult task for many people, who have been taught facts for decades, to accept that objective knowledge is hard to come by. But everyone understands the value and properties of a crude model.
Sorry, I wasn't clear. I meant that when I see a p-value of 10^-6 in papers, I expect that there's at least a 1% chance that the humans screwed up the models somewhere, so I don't see it as more persuasive than 10^-2. That is, p-values lose credibility once they start getting smaller than a few percent.
So I agree with you. If it were up to me, we'd all report evidence intensity in decibels, anyway.
It is a difficult task for many people, who have been taught facts for decades, to accept that objective knowledge is hard to come by. But everyone understands the value and properties of a crude model.