Gödel claimed that what the Theorems do entail (specifically, the Second Theorem) is that mathematics is inexhaustible:
It is this theorem [i.e., the Second Theorem] which makes the incompletability of mathematics particularly evident. For, it makes it impossible that someone should set up a certain well-defined system of axioms and rules and consistently make the following assertion about it: All of these axioms and rules I perceive (with mathematical certitude) to be correct, and moreover I believe that they contain all of mathematics. If someone makes such a statement he contradicts himself. In the Gibbs Lecture, thus, Gödel acknowledged that [his theorems] do not rule out the existence of an algorithmic procedure (a computing machine, an automated theorem prover) equivalent to the mind in the relevant sense [...]. However, if such a procedure existed “we could never know with mathematical certainty that all the propositions it produce[d were] correct.” Consequently, it may well be the case that “the human mind (in the realm of pure mathematics) [is] equivalent to a finite machine that … is unable to understand completely its own functioning”: a machine too complex to analyze itself up to the point of establishing the correctness of its own procedures. Gödel inferred that what follows from the incompleteness results is, at most, a disjunctive conclusion:
Either mathematics is incompletable in this sense, that its evident axioms can never be comprised in a finite rule, that is to say, the human mind (even within the real of pure mathematics) infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable Diophantine problems of the type specified… It is this mathematically established fact which seems to me of great philosophical interest. In other words, either the mind actually has a non-algorithmic and not fully “mechanizable” nature, or else there exist absolutely undecidable mathematical problems. But [Gödel's Theorems] don’t allow us to go further and conclude that the true disjunct is the first one. According to Gödel, then, what follows from [them], and especially from [the Second one], is that if our mind is a computing machine, it is one such that it “is unable to understand completely its own functioning.”
While the inference about disjunction is reasonable, I am unable to understand that why Godel didn't reject possibility of algorithmic nature of mind. It is always reasonable to argue that because human developed, say, an engine, his mind is more powerful than the internal logic of engine. Godel took particular interest in Turing's analysis because the result allowed Incompleteness Theorem to hold in full generality by providing the required understanding of what a "reasonable system" is.
Fact F: Turing (mind) developed the notion of machines (including the infinite -Universal Turing machine).
So my question is why was the fact F not sufficient for Godel to solve the disjunction and conclude that the human mind is strictly more powerful than a finite machine?