Yet when the Biden group seemed close to a deal, it was shot down by the tea party's champions. Boehner left Cantor exposed as the front man in the Biden talks and did little to rescue him.
Then it was Boehner's turn on the firing line. He came near a bigger budget deal with President Obama, but the same right-wing rejectionists blew this up, too. Cantor evened the score by serving as a spokesman for Republicans opposed to any tax increase of any kind.
Think about the underlying dynamic here. The evidence suggests that both Boehner and Cantor understand the peril of the game their Republican colleagues are playing. They know we are closer than we think to having the credit rating of the United States downgraded. This may happen before Aug. 2, the date everyone is using as the deadline for action. We have less time than we think.
Unfortunately, neither of the two House leaders seems in a position to tell the obstreperous right that it is flatly and dangerously wrong when it claims that default is of little consequence. Rarely has a congressional leadership seemed so powerless.
Compare the impasse Boehner and Cantor are in with the aggressive maneuvering of Senate Republican leader Mitch McConnell. He knows how damaging default would be and is working with Majority Leader Harry Reid to concoct a way out.
McConnell can do this because he doesn't confront the tea-party problem that so bedevils Boehner and Cantor. Many of the tea party's Senate candidates - Sharron Angle in Nevada, Christine O'Donnell in Delaware, and Joe Miller in Alaska - lost in 2010. Boehner and Cantor, by contrast, owe their majority in part to tea-party supporters. McConnell has a certain freedom to govern that his House leadership colleagues do not.
And this is why Republicans are going to have to shake themselves loose from the tea party. Quite simply, the tea party's legions are not interested in governing. They believe that because the Republicans won one house of Congress in one election, they have a mandate to do whatever the right wing wants. A Democratic president and Senate - also elected - are dismissed as irrelevant nuisances.
The tea party lives in an intellectual bubble where the answers to every problem lie in books by F.A. Hayek, Glenn Beck, or Ayn Rand. Rand's antigovernment writings, regarded by her followers as modern-day scripture, are particularly instructive.
When the hero of Rand's breakthrough novel, The Fountainhead, doesn't get what he wants, he blows up a building. Rand's followers see that as gallant. So perhaps it shouldn't surprise us that blowing up our government doesn't seem to be a big deal to some of the new radical individualists in our House of Representatives.
Our country is on the edge. Our capital looks like a lunatic asylum to many of our own citizens and much of the world. We need to act right now to restore certainty by extending the debt ceiling through the end of this Congress.
Boehner and Cantor don't have time to stretch things out to appease their unappeasable members, and they should settle their issues with each other later. Nor do we have time to work through the ideas from the Gang of Six. The Gang has come forward too late with too little detail. Their suggestions should be debated seriously, not rushed through.
Republicans need to decide whether they want to be responsible conservatives or whether they will let the tea party destroy the House that Lincoln Built in a glorious explosion. Such pyrotechnics may look great to some people on the pages of a novel or in a movie, but they're rather unpleasant when experienced in real life.
E.J. Dionne is a Washington Post columnist. E-mail him at firstname.lastname@example.org.