I thought you were interested to find the tests. The tests that fail are there. MT doesn't fail all, far from it. But there are RNGs that don't fail these that MT fails that are faster. PCG was mentioned here.
The only information on PCG I can find is the author's own paper which also fails some TestU01 tests. I am much more inclined to trust an established generator than an unproved one.
PCG is a family of PRNGs. Any sensible member of the family will trivially pass TestU01 BigCrush. If you use a 32-bit PCG it's going to necessarily fail BigCrush.
PRNG design is an empirical science. We can implement a generator and throw it into a statistical battery. If one passes, we've got a constructive proof of goodness. Any of us can run BigCrush and verify the results. Unlike CSPRNGs, we don't need to sit around and wait for the results of cryptanalysis.
