What was Hilbert's view of Gödel's Incompleteness Theorems?
The reaction of Hilbert to Gödel is described in detail by Solomon Feferman in Gödel on finitism, constructivity and Hilbert’s program (2011).
Hilbert was unaffected by any of the reconsiderations of the possible limits to finitary methods in pursuit of his consistency program that had been stimulated Gödel's work. In fact, there are no communications between Hilbert and Gödel and they never met. Perhaps the second incompleteness theorem on the unprovability of consistency of a system took Hilbert by surprise. We don’t know exactly what he made of it, but we can appreciate that it might have been quite disturbing, for he had invested a great deal of thought and emotion in his finitary consistency program which became problematic as a result.
There is just one comment, of a dismissive character, that he made about it four years later in his 1934 preface to Volume I of the Grundlagen der Mathematik. This is his sole reference anywhere to Gödel or his incompleteness theorems. Hilbert writes [quote translated in the OP].
So it seems the answer to the OP's question is "No, Hilbert did not pursue the objections raised by Gödel". Other contemporaries did, as explained in Hilbert's Program (2015), but apparently not Hilbert himself.
In his (excellent) essay Hilbert's Programme, Craig Smoryński has this to say:
... a variant on what the OP quoted.
[p. 53 in the linked print]
In 1931, Hilbert was sixty-nine. According to the essay, at the same conference (in Königsberg, 1930) where Gödel briefly announced his incompleteness result (at a discussion following a talk by von Neumann on Hilbert's programme), Hilbert would give his retirement speech. He apparently did not notice Gödel's announcement then and there but was alerted to the results later. To quote Smoryński, The improptu announcement of the First Incompleteness Theorem was the big non-event of 1930. [p.49]
Some related information :
1) Volume 2 of Hilbert & Bernays, Grundlagen der Mathematik (1939) include full proofs of Gödel's 1st and 2nd Theorems (for the 2nd one, it was the first published complete proof), as well as Gentzen's concistency proof, with detailed discussion of their "impact" on the finitist standpoint.
See Wilfried Sieg & Mark Ravaglia, David Hilbert and Paul Bernays Grundlagen der Mathematik I and II : A Landmark.
2) See in : David Hilbert, Lectures on the Foundations of Arithmetic and Logic 1917-1933 (Wilfried Sieg ed - 2013), the Introduction to the Appendices, page 788-on, regarding Hilbert's lectures of the '30s (and thus, "affected" by Gödel's Theorems).
3) Assuming that the the work on Grundlagen was at least "supervised" or "agreed on" by Hilbert, we can see Paul Bernays' paper of 1967 : it seems that Hilbert carried on with his foundational project post-1930, in order to take into account Godel's works :
"One step in this direction [ enlarging of the methods of proof theory, from the original finite Standpunkt ], made by Hilbert himself, was to replace the schema of complete induction by the stronger rule later called infinite induction (“Die Grundlegung der elementaren Zahlenlehre” (1931a) and “Beweis des Tertium non datur” (1931b))." [see page 24].
About Beweis des Tertium non datur, here is Wilfried Sieg's comment :
Hilbert 1931a brings in a new technique to address syntactic completeness questions for arithmetic, whereas Hilbert 1931b formulates quite novel, but also somewhat obscure, directions for further proof-theoretic work. It is clear that they react to what Hilbert and others in his School knew at the time of Gödel’s Incompleteness Theorems, and are important at the very least on that account. The last paper is also significant since it influenced Gentzen’s early attempt, starting in late 1931, to establish the consistency of full elementary number theory.
Sieg's introductions are expanded into :
- Wilfried Sieg, Hilbert's Programs and Beyond (2013) : II Analyses : Historical, page 129-on.