How is CLT related to the condition of data (normality assumption)? I apply statistical methods in sociological researches and now I feel a bit confused as I found out more about CLT.
For instance, if I have sample of 1000 observations, do I even need to check it for normality as CLT states that if sample is big enough, our data approximates normal distribution? And if it holds up, why people constantly plot q-qplot or do tests like shapiro to test for normality? (Plus we know that many of stat methods such as ANOVA, t test quite robust to violation of normality)
Of course, when population is normally distributed and we have big enough sample we can be sure that our sample is normally distributed, but what if it is not, but I still have big sample(e.g. 1000), should I really worry about the fact that my data violates this assumption?
Sorry if this is stupid question. Thank you.


CLT states that if sample is big enough, our data approximates normal distribution

That is false.
If you take a very large sample from a non-normally distributed population, the empirical distribution of the sample has high probability of being close to that of the population, which is not normal. It does not in any way converge to the normal distribution.
CLT says that the sampling distribution of the sample mean, or of the sample sum, approaches the normal distribution as the sample size grows. That is quite a different matter.
As for $1000$ being a big enough sample to get a good approximation to the normal distribution, that is not true of very skewed population distributions. For example, if $X_1,X_2,X_3,\ldots\sim\operatorname{Poisson}(0.001)$ then $X_1+\cdots+X_{1000}\sim\operatorname{Poisson}(1),$ and that is not close to the normal distribution.
