KS test with estimating parameters in t-Student case

I have some doubt:

practically my dataset, graphically speaking, seem generated from t-student o similar distribution. If I estimate Normal on the data, also graphically, the fit is no good and proceeding with standard KS test (with ML parameters) the null hypothesis of Normal, not surprisingly, is strongly rejected (p-val close to zero).

If I repeat the same strategy with t-student, that graphically speaking seem enough good fit for the data, the null hypothesis of t-student, (surprisingly or not), is clearly not rejected (p-val close to 0,28).

However if I proceed with Monte Carlo simulation as suggest in this topic:

Testing whether data follows T-Distribution

the null hypothesis of t-student (surprisingly or not) is rejected (p-val close to 0,005). (n = 3960 m=1.000).

About this problem I have a theoterical question. In what sense the standard KS test is wrong if I use it as goodness of fit strategy when the parameters are estimate from the data?

More clearly if I test the F(tetha) distribution for the data and tetha vector are known the standard KS test is applicable and I obtain as p-value in the form p=Prob(F(tetha)|data). However if I estimate tetha as tetha* seem that I obtain p-value in the form p=Prob(F(tetha*)|data) that maybe is interpretable as certain max p-value, or no?. Maybe tha fact that tetha* is estimate from the data is a problem … but theoretically speaking I can define F(tetha*) a priori without see the data and proceed with the test. What me say similar statistic? ... Is only non sense ? I think no.

Furthermore when the sample size increase, in my case is 3960, the KS standard test converge with Monte Carlo, as ML estimate theoretically converge to the true parameters, or no ?

What do you think ?