@CliffAB I agree with your comment, apparently the author of the article I referred in my question meant that LRT is for some reason good even if it is not a UMP test, and I hope that this reason is not just the ease of implementation or the lack of other alternatives. I suspect (hope) that the LRT has some good asymptotic properties (e.g. it is consistent, i.e. its power for any $H1$ goes to $1$ if we increase the number of observations).

after some time spent with probability books I see the mathematical background of the proposed solution, now I guess I can accept the answer.

I tried to put the intuition you gave to me in the previous comment in more rigorous terms here. Do you mind to take a look and let me know whether I got you right?

Thanks for the answer, I am not getting how one can prove that "the marginal ("unconditional") law of $Z_3$ also is the uniform distribution".

@Xi'an Yes, I guess it is another way to say that $X_i$s are linked through a Gaussian Copula (a more precise way).