MT only fails LinearComp systematically. It is not a bad generator, it is a very good generator. The biggest problem is the "escape from zero-land" stuff where it's biased towards zero for a while if its seed is mostly zeros, but other generators have the same problem (though not to the same extent) and it's still statistically very unlikely to be a problem and it self corrects as the sequence runs.
Comparing to ChaCha20, which is a CSPRNG, is a little apples/oranges.
Just familiarity. Mostly I just can't recommend something I don't know much about. The SIMD variant of MT is architecture specific, I thought? Other newer variants of MT/WELL generators and PCG are probably all good options. Also xorshift*.
But I know that MT is a very good algorithm and using it has a certain pragmatic appeal - you make a lot of people's lives easier because when they Google to learn more about your generator they'll find good information quickly.
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.
Wikipedia entry for MT specifies some of the problems.