Agree with you on all three, but specifically for 1., can you think of pathological pairs of distinct distribution that the test would often fail to reject?
The article says it's poor at detecting differences in the tails and much better at differences in the medians. So that's where I'd start to find problems.
Playing with the tails make all kind of mistakes possible, but that seems like a criticism that would apply to any attempt to identify a distribution based on sample.