It's very hard to believe Godel didn't face problems you faced. He probably did mistakes too, and went back periodically to make sure his proofs are sane. He was probably stuck too, many times even, and solved those problems as other mathematicians solve their problems. After all, it took him many years to complete his proof. And even more years until theorems stabilized to today's form (i.e. Rosser's trick etc). People have been practicing formal mathematics long before Godel, and this cycle of being stuck and solving problems is pretty common in mathematics. His coworkers would have had the same sort of issue. This seems to indicate me that GP comment is right that the inciting incident was understanding the correspondence between mathematics and its encoding. Which brings us to:
> E.g. Hilbert understood mathematics as a symbolic game
Hilbert had a radically different understanding of the nature of mathematics compared to Godel. Moreover, Hilbert was radical enough that he tried to use his fame to silence people with differing views than him (e.g. Brouwer). Hilbert hardly believed Godel when he published his results, because it was easy to see that it would shake the foundations of "Hilbert's Program" something Hilbert spent half his life on (and logicians still publish papers about the relationship between Godel's Incompleteness theorems and Hilbert's Program). This makes Hilbert orders of magnitude less motivated to think about this issue, probably. On the other hand, Godel had various non-mathematical reasons to prove incompleteness, one being his desire to block materialism; check for Godel's Gibbs Lectures.
> E.g. Hilbert understood mathematics as a symbolic game
Hilbert had a radically different understanding of the nature of mathematics compared to Godel. Moreover, Hilbert was radical enough that he tried to use his fame to silence people with differing views than him (e.g. Brouwer). Hilbert hardly believed Godel when he published his results, because it was easy to see that it would shake the foundations of "Hilbert's Program" something Hilbert spent half his life on (and logicians still publish papers about the relationship between Godel's Incompleteness theorems and Hilbert's Program). This makes Hilbert orders of magnitude less motivated to think about this issue, probably. On the other hand, Godel had various non-mathematical reasons to prove incompleteness, one being his desire to block materialism; check for Godel's Gibbs Lectures.