Should the t-statistic (not data) be normally distributed for using the t-test? Reading the introduction of English Wikipedia's article about the t-test, I was confused by this statement:

It is most commonly applied when the test statistic would follow a
normal distribution

To this point I was sure that the key assumption for using t-test is that the data itself needs to be distributed normally (at least approximately). But here it's stated that the test statistic should follow a normal distribution. What am I missing?
I understand that Wikipedia is not the most reliable source of information, but nevertheless I want to somehow interpret and double-check what I've read. Any help would be appreciated!

The $t$-statistic should be $t$- distributed.
This is guaranteed by math when the data are $iid$ normal, which is the legendary normality assumption. However, the t-stats often have close to the correct distribution, especially in large sample sizes, even when the data violate the normality assumption. That is, the usual t-test is fairly robust to violations of the normality assumption, and it is not so ridiculous to t-test data that clearly did not come from normal distributions (though there may be even better approaches).
