Godel's Incompleteness Theorem

I often hear people sharing this argument, and it has always bothered me...

The argument goes something like this: Godel's Incompleteness Theorem states that there exists no algorithm that can prove all mathematical theorems, there will always be some theorems that the algorithm cannot give a conclusive answer on. Humans have proved a lot of theorems; therefore, humans must possess something non-algorithmic (some make the jump to consciousness) that allows them to (eventually) prove (or disprove) any mathematical theorem.

The argument bothers me because it might very well be that we do follow an algorithm (however complex it is), and so far we have only solved algorithmically-provable theorems; and some of the theorems/conjectures we're trying to prove right now might be just out of our grasp.

Am I missing something?

This is almost surely a very basic thought, but I never got the chance to share it with someone so thought of doing that here (my first HN post :D)