It is wrong that the distribution of the data/population approaches a normal distribution.

But you are right that the sample distribution of the test statistic, which is used to estimate parameters of the population distribution or estimate the error/variance, will often approach a normal distribution independent from the underlying distribution of the population.

So for large sample sizes the violation of the assumption that the error distribution is a normal distribution becomes less of a problem. (Ironically the test of normality becomes more powerful and likely to reject the hypothesis of the assumption)

do I even need to check it for normality as CLT states that if sample is big enough, our data approximates normal distribution?

• It is wrong that, if the sample is big enough the distribution of the data/population approaches a normal distribution.

  Instead, the CLT relates to (the limit of) the mean of samples (or other types of sums of variables).

• But you are right that the sample distribution of the test statistic, which is used to estimate parameters of the population distribution or estimate the error/variance, will often approach a normal distribution independent from the underlying distribution of the population.

So for large sample sizes the violation of the assumption that the error distribution is a normal distribution becomes less of a problem. (Ironically the test of normality becomes more powerful and likely to reject the hypothesis of the assumption)