The t-test is often used in hypothesis testing when the sample size is small (less than 30) because its parameterization by degrees of freedom allows the greater uncertainty to be accounted for. Many online information sources, however, including answers in Cross Validated, say t-tests and z-tests require approximate normality in the underlying population or random variable. This Wikipedia section says that for a one-sample t-test, the underlying population or random variable does not need to be normal if the sample is large enough that sample mean is normally distributed due to the Central Limit Theorem (CLT).
Would it be fair to say that the t-test can be used: (i) when sample sizes are small or (ii) when the underlying population or random variable is not normal, but not both (i) and (ii) at the same time? That is, the above "or" should be an "exclusive or"?
P.S. I did not mention that the t-test is used when we do not know the standard deviation of the underlying population or random variable. But the above still applies, i.e., the t-test is inappropriate if conditions (i) and (ii) both apply.