Surely, ONE non-provable truth would be enough to distinguish oneself from a well-progranned IBM 360(remember those?).
You have to keep in mind that the truth will be non-provable within a specified formal system (of sufficient strength to express integer arithmetic with the operations of addition and multiplication). By augmenting that system with additional axioms, the formerly unprovable truth becomes provable. But (and here's where Gödel's insight really cuts) within the newly augmented system, another unprovable but true proposition will exist; and so on ad infinitum.
Hence, to make the argument that Gödel wanted to make, the argument that Yourgrau summarizes, one must suppose it possible that mathematical truth in its entirety might somehow be graspable, and so the human mind might transcend all formal systems, formal systems which are forever showing themselves to contain true but unprovable propositions.
I defer to your more comprehensive and current understanding of Goedel.
I had always read him to be making a more modest assertion.
'One-off' mathematical truths, so to speak, were possible. ;^)
Which violates Gödel's Incompleteness Theorems, as there no way to know if a Gödel statment is true or not without supplemental axioms, which creates a new set of unproveably true Gödel statements.... and so on, as you pointed out.
In effect, the argument being made is a case of Begging the Question, because it requires you to assume that which contradicts Gödel's Theorems (that we can can somehow prove true a Gödel statement with out creating more Gödel statements) order to arrive a conclusion (that we can grasp Mathematical truth in its entirety) that violates Gödel's Theorems.