The real difference between Z-Test (Gauss Test) and T-Test is not the population size but the assumption about the standard deviation.

In the Z-Test, you assume that you know exactly what the real standard deviation is (which is very unrealistically of course). In the T-Test you assume that you don’t know it so you have to estimate it first which is a more realistic scenario. This is the reason why the T-Test is used in the real world and not the Z-Test.