What conservatives are trying to do in Washington, what Tea Party members want them to do, is not really to cut spending. It is to wield power, and thereby to bring self-described conservatives back into power. Part of why American conservatives were so angry in 2010 was that they were shut out of power, and Democrats were able to enact a big new government program (Obamacare) despite their best efforts to stop it. Whether Obamacare is a good idea or not is beside the point; most Tea Party voters have no idea what is even in the bill. (Neither do many of the bill's Democratic supporters.) The point is that it was a liberal plan that was shoved down their throats.
The job of Boehner and his comrades now is to force Obama and the Democrats to acknowledge their power, and therefore the power of American conservatism. They say, although I don't believe them, that they are willing to let the government hit its debt ceiling and default on its financial obligations to achieve that. To force their will on the country, and therefore validate their centrality in America, conservatives are willing to do something really stupid and quite radical.
And that is Democracy.