Bayesian criticism of frequentist p-value I recently started reading the Bayesian criticism of the p-value and it seems that there is a lot of discussion around the fact that a frequentist approach is not that good when the Null Hypothesis is true.
For instance in this paper the authors write that "p-values overstate the evidence against the null [...] this does not have to do with type-I or type-II errors; it is an “independent” property of p-value."
To illustrate this point, the authors show that when the null is true, the p-value has a uniform distribution.
What I do not get is that even when the null is true, a frequentist approach, thanks to the Central Limit Theorem, is still able to construct confidence intervals that includes 0 (non-significance) at the appropriate $\alpha$ level.
I do not get why the fact that the p-value is uniform when the null is true shows that a frequentist approach is biased. And what does it mean "independent property of p-value"?

library(tidyverse)
library(broom)

n=1000
x = rnorm(n,100,30)
d = 0
y = x*d + rnorm(n,0,20)
df = data.frame(y,x)
plot(x,y)
abline(lm(y~x), col = 'red')

r = replicate(1000, sample_n(df, size = 50), simplify = F)
m = r %>% map(~ lm(y~x,data = .)) %>% map(tidy)

# Central Limit Theorem
bind_rows(.id = 'sample', m) %>% filter(term =='x') %>% ggplot(aes(estimate)) + facet_grid(~term) + geom_histogram()

s = bind_rows(.id = 'sample', m) %>% filter(term =='x')
s$false_positive = ifelse(s$p.value < 0.05, 1, 0)
prop.table(table(s$false_positive))

# uniform
hist(s$p.value, breaks = 50)


The discussion here is excellent but at the heart of the matter is that in an attempt to "let the data speak for themselves", i.e., to be objective, the frequentist approach jettisons the desire to obtain measures of evidence in favor of assertions.  As explained eloquently in Bernoulli's Fallacy, Clayton takes us step by step through statistical history to explain how this happened and what harm it has done.  One of the harms is that outside information was prevented from being brought into the analysis.  One of his excellent examples is ESP research where he shows that obtaining a low p-value is meaningless without factoring in the low likelihood that the laws of physics are allowed to be suspended.  So one can analyze all day the amount of evidence in a p-value but I don't think that is quite worth the trouble.
