I’m sorry, but I don’t agree with the t test definition. It’s not supposed to tell you if two distributions are equal. It’s supposed to tell you if two means are similar (or if a mean value is similar to a known expected value). Concerning the normality assumption, I’ve always had some doubts. The mean value of N independent, finite variance random variables converges to a normal distribution, so the normality assumption is quite satisfied. If the sample is too small, we can’t even talk about normality. That’s why Gosset introduced the t distribution. It can be considered as a finite-size version of the normal distribution.