Most studies use raw p-values. This, coupled with "exploratory analyses" which are not mentioned in the research, leads to a high false discovery rate.
Alternatives include using simulations to estimate false discovery rates (this can be done analytically for some problems). Bayesian frameworks can also be applied, depending on the amount and accuracy of prior knowledge of the system under study.
That said, I see nothing wrong with publishing results with nominal p-values, as long as the researchers indicate the weakness in their results. Meta-analyses can always come back later and use the results they publish.
What annoys me is seeing 50 or so statistical tests done, and then researchers stating they have found something when 1 of those tests shows a p-value of 0.05. Just using a Bonferroni correction, the simplest of all corrections, would demonstrate that findings like this are not significant.
The problem isn't with the use of p-values, per se: the problem is that the human brain tends to distort probabilities on either end of the spectrum. People see that there's only a 5% chance that something could have happened by chance, and they put that in the "Not possible" bucket.
It's a fundamental weakness of the human brain. There's not a whole lot we can do about it.
Hypothesis testing isn't the fundamental problem, it's how a given p-value is perceived. Getting a so-called significant result simply shouldn't be confused with conclusive evidence for the hypothesis.
