Under null hypothesis, p-values are uniformly distributed

Central limit theorem states that sample means follow normal distribution. People often confuse this with and question the validity of uniform distribution of p-values under nul hypothesis. Should we not be observing extreme p-values much less ofter than usual pvalues like 0.2 or 0.3?

Imagine if null hypothesis is true. This means there is no difference betweem treatment and control.

import numpy as np
import pandas as pd
import statsmodels.api as sm
from scipy import stats
import matplotlib.pyplot as plt

pval_lis = []
t_stat_lis = []
samp_mean_lis = []

for i in range(100000):
    sample = np.random.normal(0, 1, 10)
    samp_mean = sample.mean()
    samp_mean_lis.append(samp_mean)
    t_stat = sample.mean()/(sample.std()/np.sqrt(sample.size))
    t_stat_lis.append(t_stat)
    pval = stats.t.sf(abs(t_stat), df=sample.size)*2
    pval_lis.append(pval)

plt.hist(np.array(samp_mean_lis))

Histogram of sample means — Distribution of sample means

plt.hist(np.array(t_stat_lis))

Histogram of t-statistics — Distribution of t-statistics

plt.hist(np.array(pval_lis))

Histogram of p-values under the null hypothesis — Distribution of p-values under the null hypothesis — uniformly distributed