Is Goedel's 1st theorem not algorithmically derivable? First let me explain what I mean by algorithmically derivable.
An algorithm must be able to come up with the proof without prior knowledge of the proof, in the same way mathematicians and computer scientists are able to create new proofs without prior knowledge of the proofs.

Our proof deriver enumerates all possible axiomatic systems, and with each system enumerates all possible proofs.
It checks each proof to see if it proves Goedel's 1st theorem.
This algorithm will necessarily derive all finite proofs eventually.

Goedel's 1st theorem states no sufficiently powerful axiomatic system can be complete and consistent at the same time.
The deriver is complete, since it will eventually be able to prove all true statements and disprove all false statements.
This means the deriver is inconsistent, so it cannot know whether a set of axioms used to prove Goedel's 1st are consistent.

Therefore, an algorithmic deriver is unable to derive the proof for Goedel's 1st theorem.

Does this proof succeed in showing Goedel's 1st theorem is not algorithmically derivable?
Your reasoning is incorrect.

It is true that your hypothetical "proof deriver" cannot derive all true statements. No proof derivation system can, and indeed, it is not even possible to express the set of true statements in arithmetic, which is a consequence of Tarski's theorem on truth, itself a consequence of Gödel's theorem.

However, your algorithm does enumerate all provable statements, including the incompleteness theorem itself.

In fact, here's a much simpler algorithm:


Print a formal proof of the incompleteness theorem.
Halt.


So you need a much subtler argument to prove that computers can't do what humans can do in terms of proofs. Certainly you would need a better notion of what "prior knowledge" of a proof means. Indeed, many people today believe that there are no tasks which are fundamentally impossible for computers but possible for humans, and I know of no unassailable argument for that position.