I agree that the system can't use this argument, but that doesn't mean the argument doesn't work, just like your argument for why your algorithm must defect against a program that always defects.

The difference between my argument and "this statement is false" is that by directly depending on its own truth, "this statement is false" leaves its truth or falsity insufficiently defined. But "this statement is true iff it is provable" depends not directly on its truth but on provability, which is already defined. So I think that my "real world" argument does establish that the statement will be true also within a formal system, even though a formal system cannot establish it with this argument.