The clearest, least mystical, presentation of Goedel's First Incompleteness Theorem is: nonstandard models of first-order arithmetic exist, in which Goedel Sentences are false. The corresponding statement of Goedel's Second Incompleteness Theorem follows: nonstandard models of first-order arithmetic, which are inconsistent, exist. To capture only the consistent standard models of first-order arithmetic, you need to specify the additional axiom "First-order arithmetic is consistent", and so on up the ordinal hierarchy.

This doesn't make sense. A theory is inconsistent if and only if it has no models. I don't know what you mean by an "inconsistent model" here.

Now consider ordinal logic as started in Turing's PhD thesis, which starts with ordinary first-order logic and extends it with axioms saying "First-order logic is consistent", "First-order logic extended with the previous axiom is consistent", all the way up to the limiting countable infinity Omega (and then, I believe but haven't checked, further into the transfinite ordinals).

Actually, it stops at omega+1! Except there's not a unique way of doing omega+1, it depends on how exactly you encoded the omega. (Note: This is not something I have actually taken the time to understand beyond what's written there at all.)