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.

I think that's a poor characterization of the purpose of mathematical logic. The purpose of mathematical logic is not to capture the human mind, it's to overcome the limitations of the human mind.

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.

Did it succeed in that regard? Gödel said that first-order logic powerful enough to capture Peano Arithmetic does not have a sound proof theory (incompleteness = not onto = unsound), meaning human imagination is still the limiting reactant.

Yes, it did succeed in that regard.

To use your example, Godel would not have been able to explore those ideas without using mathematical logic.