[Question] How important are assumptions in hypothesis tests?

Certain statistical tests, such as the Z-test for an equality of a mean, chi squared test for cont. tables and the significance of the correlation coefficient are often based on certain assumptions, such as data that is normally distributed. However, often i seem to not see any visual description of the data that is being tested (for example histograms) or any tests (like the Kolmoforov-Smirnov test) being showcased for the distribution of the data. I understand that the test assumptions might be sattisfied or differ insignificantly when the data follows a distribution similar to a normal one,