No. The product-moment correlation coefficient r doesn't have to be tested with a t statistic (or an equivalent like F). Use of t stems from the assumption of bivariate normality for the test ... and is not intrinsic to r. For example you can bootstrap tests for r and this is recommended if the assumptions of the test are violated.

You could also motivate approximate asymptotic tests using the z or equivalently chi-square distribution (but that would be inferior to using t or F).

Or do Bayesian parameter estimation and get the posterior distribution of r.

I was also interested in this matter, look at

Method Visualization of Correlation Tables by Positive/Negative Thr...