Usually, "what happens" is that the machines become large enough to represent a form of induction too strong for the axioms to 'reason' about. It's a function of the axioms of your theory, and you can add more axioms to stave it off, but of course you can't prove that your new axioms are consistent without even more axioms.

> There is a presentation of it that is in like less than one page in terms of the halting problem---all of the Gödel-numbering stuff is essentially an antiquated proof.

Only insofar as you can put faith into the Church–Turing thesis to sort out all the technicalities of enumerating and verifying proofs. There still must be an encoding, just not the usual Gödel numbering.

> There is a presentation of it that is in like less than one page in terms of the halting problem

Those are two very different ideas. Your second sentence says that Gödel's theorem is easy to prove if you have results about the halting problem. Your first one says that in order to prove Gödel's theorem, you need to establish results about the halting problem.