# Why does test on Pearson correlation require bivariate normality?

For a pair of random variables $X$ and $Y$, we can compute their Pearson correlation coefficient $r$ and conduct hypothesis testing on the null hypothesis $H_{0}:r=0$ with the $t$ statistic
$t=r\sqrt{\frac{n-2}{1-r^{2}}},$ which follows a $t$ distribution with $n-2$ degrees of freedom.

I understand how to derive the distribution of this statistic by considering it an OLS regression, i.e.,
$X=aY+e_{x}$ or $Y=bX+e_{y}$

However, in OLS regression, I think we only require the error term to be normally distributed, i.e.,
$e_{x}$ ~ $N$ and $e_{y}$ ~ $N$, which implies that $X|Y$ ~ $N$ and $Y|X$ ~ $N$

Thus, it seems that in order for the $t$ statistic to follow the desired $t$ distribution, only the two conditional normalities (i.e., $X|Y$ ~ $N$ and $Y|X$ ~ $N$) are required.

However, according to the wiki page, joint normality for $X$ and $Y$ is required, so my questions are
(1) why is the bivariate normality required?
(2) in contrast to OLS regression where only conditional normality is required, is it true that the testing for $r$ also requires marginal normality (i.e., $X$ ~ $N$ and $Y$ ~ $N$)?