The argument against p-values is part of the argument against any bright-line single-number rule for identifying truth. The job of the researcher is to demonstrate (at least) that
1. there is (isn't) an observable difference between groups of interest
2. the difference is (not) attributable to hypothesized causal mechanism i.e. the (absence of a) difference isn't due to random variation in the observed sample i.e. the difference would be observed by a independent replication of the same analysis/study
3a. the difference is not explainable by other factors that vary between the groups, observed or unobserved
3b. the difference is not artificially inflated (suppressed) by the statistical choices
4. the difference is large enough to be practically relevant.
and so on
If the degree of certainty of statements about the difference can be characterized by a single number at the end of the process, great! But the goal should be a convincing, wholistic story, not the single number.
I share your concern, and I worry that we'll find this battle continues 20 years from now.
There are many possible things that can go wrong with a P-value and I'm not a statistician, but things I look for in data are the structure/distribution of the "noise" and any correlations seen within the "noise" and the "signal". That helps you build a signal and noise model. Assuming that all your noise is inherently uncorrelated gaussian, is a pretty strong model assumption.
So, what’s the best way to measure the probability that the studied hypothesis is true?