So, assuming the initial system is consistent, adding the godel sentence, or its negation, produces a new consistent system, iirc. But in the second case, it would be omega-inconsistent ?

Assuming the original system was consistent (which you can never prove), yes.

But that doesn't help you, because the new system with these additional axioms will still, provably, contain statements that can't be proven true or false within that system (or else be inconsistent). (Turing did try some programme where you keep adding new axioms and take the transitive closure over all the axioms you'd add, but AIUI that doesn't work out either).

Careful there: Gödels first incompleteness theorem proves that in a consistent effectively axiomatized formal system, there will be statements that are true or false that cannot be proven to be true or false.

So let's assume there were such a thing as objective mathematical reality and that it could be formalized as The System. As far as I'm aware, all the incompleteness theorem implies is that The System's set of theorems is not recursively enumerable (ie we will never be able to 'write them down').

All we can do is craft necessarily incomplete subsystems applicable to areas of our interest. It's a bit sad that humans will never know the Whole Truth, but such is life...