Where does the argument "Pearson's correlation coefficient assumes normality" come from?

Question

Related1: Pearson's or Spearman's correlation with non-normal data

Related2: How robust is Pearson's correlation coefficient to violations of normality?

Related3: Why is Pearson's ρ only an exhaustive measure of association if the joint distribution is multivariate normal?

I've seen a claim "Pearson's correlation coefficient assumes normality" sometimes. The three answers above explain why the population needs to be normal - the summary is that:

・the variates may have some association other than the correlation, but the Pearson's correration coeff can't tell it anyway, thus it's exhaustive only when bivariate normal is the case.

・the range gets narrower than -1 ~ +1 when bivariate normal is not the case. Especially, the accepted answer in Related2 demonstrates that normal vs lognormal, for example, can get neither -1 nor +1 at its best.

However, uniform vs uniform, for example, apparently can achieve the -1 or +1 coefficient. A trivial demo with Python program:

# This is virtually the same as the demo in Related2
# except here uniform vs uniform is used.
import numpy as np
xs = np.random.uniform(2, 5, size=10**5)
ys = np.random.uniform(1, 2, size=10**5)
xs.sort()
ys.sort()
np.corrcoef(xs, ys)[0, 1]
# 0.99998421283590666

And now I'm completely lost. Where is the origin of "Pearson's correlation assumes normality" and what is the justification?

Answer 1

I also think that the assumption of normality concerns only the calculation of a P-value of the correlation coefficient. If the variables are not normally distributed then perhaps the best thing is to calculate the P-value by the means of a permutation procedure.