Thats not really true though. The vast majority of t-tests are done using a sample average as a test statistic, and by the CLT, as the sample size goes up, the

Thats not really true though. The vast majority of t-tests are done using a sample average as a test statistic, and by the CLT, as the sample size goes up, the distribution of a sample average becomes approximately normal. So unless you have a really weak sample size, the t-test assumption holds even in the absence of normally distributed data.