You are correct that Goedel really isn't hard. And, of course, its not a problem but a theory.

The thing is Goedel shaped the development of logic itself tremendously. Before Goedel, the idea was that proven theorems were essentially the true theorems. The concept of theory proving as a simply mechanical process wasn't there, the idea of models and true-but-not-provable theories wasn't there etc. Contrawise, once you have this machinery, the theorem is fairly simple (if sometimes tedious) to produce.

So sketching Goedel's theorem can't help but be sketching early twentieth century logic.