If the goal of formal mathematical logic is to capture the subconscious patterns of thinking humans use, then by that standard, it is a colossal failure.
Math is like sailing a ship during the age of sail. In the middle of the ocean, there is no point of reference. You have a vague sense of direction, but it's easy to get lost. You need some tools to navigate. Math is the ocean, your mind is the ship and mathematical logic is the navigation instruments.
To use your example, Godel would not have been able to explore those ideas without using mathematical logic.