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?

There is some literature comparing the power of different normality tests, often involving both specific tests for normality and more general goodness-of-fit approaches that can be applied to general distributional shapes (so as pointed out already, only calling the former "normality tests" is a misnomer and many would also include specific normality tests in the more general class of "goodness-of-fit tests").
The power generally depends on against what kind of distributions normality is tested, however pretty much all that I have seen favours Shapiro-Wilks over Kolmogorov-Smirnov and Chi-Squared.
See for example
Thode HJ. Testing for normality. New York: Marcel Dekker; 2002.
B. W. Yap & C. H. Sim (2011) Comparisons of various types of normality tests, Journal of Statistical Computation and Simulation, 81:12, 2141-2155, DOI: 10.1080/00949655.2010.520163
Googling "compare normality test" or looking into the references of those cited above will bring up more.
