Why use normality tests if we have goodness-of-fit tests? What are the reason/s to use a nonparametric normality test (e.gr., Shapiro-Wilk, Jarque-Bera) instead of generic, parametric goodness-of-fit tests (good for any distribution including but not limited to the normal, with parameters, like $\chi^2$ or Kolmogorov-Smirnov) for some data we want to check for normality?

First, it's worth noting that testing for normality is a basically useless activity (cf., Is normality testing 'essentially useless'?).  No dataset in the real world is normally distributed, so we already know the null hypothesis behind these tests is false.  What's left is that the test can correctly reject the null, if the sample size is large enough relative to the way the data deviate from true normality, or can yield a type II error, if the dataset is relatively smaller.  However, what really matters isn't how many data you have, but the size and nature of the deviation from normality, which tests can't tell you.
That having been said, the reason specialized tests like the Shapiro-Wilk are used instead of generic goodness of fit tests, is because we primarily care about some specific types of deviations from normality.  Data can deviate from normality in potentially innumerable ways.  For simplicity, you can imagine a distribution that has the same kurtosis (fat-tailed-ness) as a normal, but differs in being skewed, or a distribution that differs in kurtosis, but is perfectly symmetrical.  If you tested one of those parameters, you would miss the other.  Of course, a general test will in some sense cover everything, but not with equal power—it will be more sensitive to some deviations than others.  Which deviation will be most detectable will differ by test.  Thus, you might as well use the test that is maximally sensitive to the deviations you care about.  Those are deviations in the tails, and the Shapiro-Wilk is weighted to preferentially detect them.
