First of all, I'd like to agree with simonm, I'm just continuing the next part because I think your logic is incorrect.

the other way around, I'm saying the statement is false because UTM says it is true.
Why does the UTM say that the statement is true? You have asserted this several times, but as a universal truth machine, it is unable to utter a falsehood. While the statement is true, if the UTM attempted to state as much it would be uttering a falsehood. Thus UTM cannot ever state that the true statement is true, even though it is.