Ok, so the big issue is that statistical tests like the chi-squared test are not designed to show that a sample matches a certain distribution. Statistical test

Ok, so the big issue is that statistical tests like the chi-squared test are not designed to show that a sample matches a certain distribution. Statistical tests are designed to show the opposite--"this sample does not match that distribution". If the sample matches the distribution, by design the p-value is going to be uniformly distributed--i.e. a p-value of 0.01 is equally likely as a p-value of 0.99.