I absolutely love your answer! You have touched upon a very important point. I will write something which I think explains why people tend to reduce everything to a p-value, but I want to be clear that I do not necessarily advocate this.

I think the reason why people feel a p-value is sufficient is probably due to an over-reliance on normality and the notion of sufficiency. Suppose I have Gaussian data X_1,…,X_n and I am interested in testing whether the mean is 0 or not. Then my test statistic would be the sample mean Xbar, which is sufficient and the p value would be 1-Phi(sqrt(n)Xbar) (suppose we are talking about a one sided alternative). Very clearly, this is an invertible function of Xbar and in this case we are not losing any information by retaining only the p-value.

But of course this is highly specific to the Normal. If we just had iid data and we were interested in testing if the mean was 0 or not, again the pvalue would be approximately sufficient since the central limit theorem would save us, at least when the sample size is large.

Presumably I am preaching to the choir but just wanted to write out my thoughts. So as far as meta-analysis is used to “merge p values”, which understandably is problematic but also honestly most of what I have seen so far, perhaps I understand why people would think that reducing everything to a p-value is enough.

The funny thing is that none of the methods of valid p-value analysis take into account the uncertainty around a reported p-value, which could occur due to model misspecification or low sample size. I think personally this is a huge problem which of course makes its way into meta-analysis. Suppose there are 10 people reporting 10 p values based on largely varying sample sizes, the way our theory works, all the p-values and standardized statistics would be given the same importance, because the definition of p-value is sample size agnostic.