How do you convince a GOP voter that "the government" includes the military, Social Security and Medicare?
Breach the debt ceiling. If that happens, even voters that like the GOP's current tactics will face a blunt reality: if by messing with "the government" the GOP ends up causing damage to Social Security, then logically Social Security must come from "the government".
Yes, the defining characteristic of American Conservatism circa 2013 is the tactic of avoiding cognitive dissonance by disregarding ones own lying eyes. But if anything can get the attention of old white people, it's messing with Social Security!
Even without a debt ceiling crisis, the government shutdown could permanently tilt the political landscape a little more liberal thru the power of "you don't know what you've got till it's gone."