I'm not a specialist in statistics, but I spoke to one who found this behaviour dubious.
Before using DistributionFitTest I was doing some tests with the normal distribution, like this
data = RandomVariate[NormalDistribution, 10000];
According to the documentation "A small p-value suggests that it is unlikely that the data came from dist", and that the test assumes the data is normally distributed
I found this result for the p-value to be really low, if I re-run the code I often get what I would expect (a number greater than 0.5) but it is not at all rare to obtain p values smaller than 0.05 and even smaller. Through multiple re-runs I notice it fluctuates by orders of magnitude.
The statistician I consulted with found this weird since the data was drawn from a a normal distribution and the sample size is big, especially because the Pearson X2 test also fluctuates like this: