Estimating proportion of real positives in the population

#1
I have a population of things, say genes. I think some of them will have a feature, say involved in metabolic syndrome. If I do a frequentist test for this feature I expect the ones that do not have this feature to produce a p value with the uniform distribution, and the ones that have this feature to produce a p value skewed towards zero.

If I am interested in selecting a subset of this population to reject the null hypothesis with a certain confidence I could do something like the Benjamini Hochberg procedure. If however I am interested in the characteristics of the whole population, such as how many have this feature or even what is their effect size distribution, what techniques could I use?

I have tried searching for this, but I really do not know what search terms to use and can find nothing. I have thought about it and can imagine assuming the true effect size follows a beta distribution in the cases where the null hypothesis is false and estimating the parameters of the distribution with monte carlo bayesian inference, perhaps with BUGS or STAN, feeding in as observations p values and effect size estimates. I feel someone must have thought about this before.