The flaw was the belief that it was possible (though they were not sure that they two could pull it off). Gödel showed that it was not an impossibly difficult task but an actually impossible task.
And there was no "flaw" in attempting to resolve foundational paradoxes in set theory, primarily to address Frege's error and deal with Russell's paradox, or in attempting to prove that all mathematics can be derived from pure logic. Seeking to demonstrate a hypothesis that is later shown to be erroneous is not "flawed", else the entire knowledge-seeking enterprise is "flawed".
And switching from "logical error" to the very vague notion of a "flaw" is goalpost moving that looks a lot like bad faith.
I won't comment further.