When one performs an ANOVA, (s)he always end up calculating the observed F-ratio and comparing it to the appropriate F-distribution. From this post, I discovered that the coefficient of correlation $r^2$ follows a beta distribution under the null hypothesis and I am wondering why we are not using the $r^2$ rather than the F-ratio to calculate the P.value.
Could we use the calculated $r^2$ and compare it to the beta distribution (make a Beta-Test) rather than comparing the F ratio to the F-distribution? If yes, do we use a F-test rather than a Beta-Test just for historical reasons or is there any other reason behind this choice? Would we systematically get the same P.value?