Roughly, Gödel demonstrated that in a logical system, that contains a model or arithmetic, there are statements which may be true, but are unprovable within the system.
If a statement is not provable an inconsistency or self-contradiction may or will develop that invalidates the system. Arrival at this point then demonstrates that a system has been considered or examined sufficiently to move on. Should arrival at this point be the focus for examination of any system?
A statement being uprovable is very different from it being an inconsistency of the theory.
If a statement is shown to lead to an inconsistency, then you can use it to prove statements and their converse statement, which cast a bad shadow on all statements in the theory. Even then, you don't want to drop the theory but examine what led to the bad apple statement being provable, and maybe delete the bad axioms.
If a statement is unprovable in some theory, but a lot of other statements are provable, then some people might think "that sucks", but this doesn't imply that there will be an incosistency. Otherwise we'd know arithmetic will be inconsistent. So arrival at this point doesn't imply at all you should stop caring about that model.
Do Godel's incompleteness theorems support the idea that the examination of a 'system' should only be undertaken to arrive at the inconsistency?
The theory of differential equations (with all its facets and connections to other topics such as topology and pure existence questions) is one which clearly incorporates arithmetic and also much bigger mathematical problems inherited from its set theoretical basis. There is still a lot to find out about differential equations and the solutions are needed in physics to build your iPod. Should the 'system' only be examined to find inconsistencies in it? Not if you want a new iPod! Why should "examination of a 'system' ... only be undertaken to arrive at the inconsistency"? Mathematics is not done for mathematics sake, by the majority of people who do it.
On that note, I find it problematic that you ask for puropose of a human activity here. The consequence of the discovery of an inconsitency is just another statement about the particular model or similar ones. "Invalid" as commonly used is a strange word in this context (not to be confused with the technical term valid in logic). Because there are several similar theories, with which you do arithmetic and arithmetic like operations. If something in a 'system' is true, say the statement "2+3=5", and the theory as a whole turns out to be invalid, I'd still use "2+3=5". Similarly, something being wrong in a particular logic or theory doesn't imply something about the validity of "the same" statement in other theories, where "validity" here really only has operational meaning.
If a statement is not provable an inconsistency or self-contradiction may or will develop that invalidates the system.
You are somehow mixing together the first and the second incompleteness theorem and drawing a misleading conclusion. Briefly put:
The first theorem proves that all consistent axiomatic formulations of number theory which include Peano arithmetic (or stronger) include undecidable propositions.
The second theorem proves that no consistent axiomatic system which includes Peano arithmetic (or stronger) can prove its own consistency.
That the axiomatic system may be inconsistent or not has nothing to do with the existence of undecidable propositions in that system. Inconsistency and incompleteness are not related in that way.
One relation that does hold though is the following: any inconsistent axiomatic system is complete, via explosion principle.
Should arrival at this point be the focus for examination of any system?
I don't see any particular reason that should be the case, nor can I see how Gödel's theorem provides any moral imperative as to how one should treat a system, or which aspects should be the subject of examination.
Personally, I think it is good that no such imperative exists-- as certain systems (say, those of mathematics) are incredibly useful, and I'd hate to see the examination of them needlessly constrained.
