It could, but doesn't need it.
That models are not direct truth is just logical. If there were such a thing that were indistinguishable from the direct truth, it would be the direct truth, not some other thing called a model.
This is why I've never understood the "revalatory" nature of Godel's incompleteness. Of course any axiomatic system must not contain all truths. Axiomatic systems are models and models can't contain all truth. If an axiomatic system contained all truth it would just be all truth rather than some other thing called an axiomatic system.
The only case in which Godel's argument surprises is the case in which one has a fundamental misunderstanding of what a model can and cannot be, which I suppose must have been more common in his time, but I was really confused when my highschool teacher went to great lengths to explain that a model cannot contain the whole truth because...of course it can't?