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.