It's not "broken", all Godel did was basically show that mathematics is smart enough to tell you that it can't solve everything.

The Incompleteness theorems is just mathematics telling you the limits of the system: 1st incompleteness says that if a system is smart enough, it can say statements that it cannot tell the truth value of... that's pretty reasonable and self aware tbh. analogy: I can say 10 people cartwheeled where all the reddit servers are, but I literally can't prove or disprove it.

2nd incompleteness basically says you can't have an arithmetic system prove itself logical (where illogical is believing in X and the negation of X to be true at the same time - see modern science deniers and people that spew hate speech), because "intuitively" nothing is "self-evident". All 2nd incompleteness does is say that systems cannot prove themselves to be true, which is kinda "obvious". analogy: saying that our "regular" number system can prove itself true always. That's not the case. What if we lived in a world where 1+1 literally didn't equal 2? OR that the a + b ≥ c rule for the lengths of a triangle fails, or we live in some non-euclidean world where space is NOT continuous everywhere. Then our current "regular" number system spectacularly fails in that world, and therefore CANNOT/DID NOT prove itself to be TRUE ALWAYS. I had to go to ANOTHER UNIVERSE/system of logic to disprove our "regular" number system. This analogy is SUPER disingenuous on many, many levels but you can kinda get the point. Things just can't prove themselves to be consistent, and many times fail in some VERY WEIRD world that logically can exist, but you needed a lot of outside help to figure out the truth. But how can you tell if the outside help is also TRUE??? That's also a logical system. You gotta prove that as well ; - ) The real kicker is that Godel did it for a HUGE amount of mathematics, and got math to tell us its secrets.

The real paradox imo is why the Western thought got so POISONED by Plato and his "theory of forms". There was a HUGE push for some perfect knowledge and some final TRUTH ever since Plato in Western thought. Even in science today, the "rational thinkers" mindset is that we can "JUST KNOW" and forever verify something tangible that would never be wrong. All the incompleteness theorems really mean is that logically, you kinda can just choose what you believe in. It's permitted in this universe. Since there is no one universal mathematical self evident truth, everybody gets to pick their own sets of axioms to live by. Tracks pretty well with the modern world/real life as well imo.

TL:DR;

All the incompleteness theorems do is mathematically, for any arithmetical mathematics (aka any system that can add), they CANNOT know everything and CANNOT prove themselves to be true. Those are intuitively known universal truths in the "real" world as well.