Here's my current theory: Obama has said that he will not sign a deal unless it extends the debt ceiling past the presidential election. The Republicans want to use this latest crisis to dismantle another large portion of the federal budget. Failing that, they're perfectly willing to have Obama ignore the debt ceiling as an exercise of the 14th amendment.
So long as the Republicans can make it seem like they made a reasonable offer, they win either way. If Obama caves like always, they win because they get to kill universal healthcare. If Obama finally mans up, they get to say that Democrats are increasing the deficit, they can say he should be impeached for breaking the law, etc., etc. The only way Obama can make this look good for him is by making it clear he went completely out of his way to make a deal, offering up Democratic sacred cows.
He did offer to increase the age for Medicare availability, (which would kill a lot of people, I would guess more than 100, but fewer than 10,000 a year) so he made the appropriate gesture. The question is whether he can sell this fact to the people of the United States. He will almost certainly win re-election against this terrible Republican field, but he needs a congress that won't threaten to blow up the country every time they don't get their way.
So this all comes to marketing, which Democrats have been incompetent at since LBJ. Hooray for the future.