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!

Quoting (the complete sentence) from Wikipedia

It is most commonly applied when the test statistic would follow a
normal distribution if the value of a scaling term in the test
statistic were known (typically, the scaling term is unknown and
therefore a nuisance parameter).

This sentence is true. Indeed, as stated in Dave's answer, the $t$-statistic follows Student's $t$ when the population variance is unknown. However, when the latter is known and you use it in place of the sample variance, the implied statistic is actually a $z$ statistic, i.e. it has distribution $N(0,1)$.
On the other hand, if the data do not come from the normal distribution and if the true distribution is not too wired, the $t$-test is still useful; see this answer by Stephan Kolassa.
