Since there is no record of Gödel's view, any response to this question is a matter of opinion.
One of the few philosophers who has published opinions on the merits of Gödel's philosophical views, as well as biographical details on role of philosophy in Gödel's professional life, is Palle Yourgrau of Brandeis University. It is his opinion, expressed in this book The Disappearance of Time: Kurt Gödel and the Idealistic Tradition in Philosophy, that Gödel did indeed consider himself to be primarily a philosopher. I've skimmed the text a number of times but I am unable to locate the paragraph where Yourgrau explicitly states that he believes this is Gödel's view of his own work, which I am certain is there. The best I can do is the following quote :
To philosophers, then and now, he was simply a logician trying to pass as a philosopher.
As Yourgrau notes, in general philosophers appear to hold a different view. Quoting Yourgrau:
Gödel, throughout his academic life, was exceptionally anxious to avoid being considered a philosophical dilettante or crank, and a "pious" one at that. In this he failed completely. His "special caution" succeeded only in keeping him from contributing to important fields of research. His refusal to publish his ontological argument for the existence of God fooled no-one. He could not hide the fact that he was a kind of believer and that his argument, like Leibniz's before him, was anything but a mere intellectual exercise. What Bertrand Russell called Gödel's' "unadulterated Platonism" marked him for some as a throwback to "precritical" times, before Kant launched is "critique" of pure reason . . . . . Even Gödel's writings on Einstein succeeded only in convincing cosmologists that strange things happen when a logician pays too much attention to the equations of relativity, while forgetting their physical meaning. His absence, in turn, from the Wittgenstein revolution . . . . . combined with his refusal to pay homage to the preeminent figure in contemporary analytic philosophy, W.V.O. Quine, marked him as a philosophical castaway.
In February of 1995, at Boston University, a symposium was held entitled "Gödel's General Philosophical Significance". The two main speakers, Warren Goldfarb and Burton Dreben, both Harvard philosophers, were unrelenting in their ridicule.
Following Goldfarb's speech, a single question was asked :
"Do I understand you correctly, professor Goldfarb, that in your judgement Gödel, though a great logician, was a philosophical fool?" A polite smile was Goldfarb's only answer.
Dreben appears to have particularly irked that Gödel had the "audacity" to engage in rational theology - no doubt a reference to Gödel's recasting of the ontological argument.
For Dreben, it seemed, this was a kind of scandal. He was moved to deliver a sermon on the harm that is done when people who are good at purely formal thinking get the idea that they are qualified to contribute to philosophy. That Dreben's own position in the philosophical world owed much to his reputation in formal logic was an irony that appeared to be lost on him.
In the years since the Boston symposium, philosophers' assessment of Gödel's philosophical work has been more polite, but no less dismissive. Yourgrau appears to be a lone voice in support of Gödel's philosophical views.
The case for Gödel as philosopher is unassailable. Though he published few essays that could be considered explicit contributions to philosophy, they suffice to establish him as an important philosopher of mathematics and time and space. The posthumous publication of several more of his philosophical studies, including the Gibbs lecture, his contribution to the Schilpp volume on Carnap, and the longer version of his contribution to the Schilpp volume on Einstein, confirm this assessment. The essay he wrote for the Schilpp volume on Russell, which contained new and insightful discussions of Frege as well as Russell on the question of meaning, including an illuminating and prescient comparison of Russell on "denoting" with Frege on "sense and reference", leaves little room for doubt that in the philosophy of language, too, his abilities were striking. He had clearly mastered the writings of most of the seminal figures in twentieth-century analytic philosophy ... His grasp of Hegel astonished the philosopher Georg Kreisel, a man not easy to impress. ( paraphrasing: He also demonstrated extensive knowledge of Kant and Husserl.)