This impasse reminds me of the many budget squabbles between President Clinton and Republicans. Even though GOP lawmakers frequently threatened to "shut down" the government, Clinton would stand firm. In the end, the GOP would capitulate, and Clinton would emerge with greater popularity. Sources say that Clinton advised Obama to use the playbook he employed against Republicans. Apparently Obama followed Clinton's advice, and it has seemingly worked. What exactly was wrong with "two presidents"?
Please bear in mind that neither side would have allowed the government to default on its debt. Instead, this was just a game of political chess. One side had to concede eventually. Lacking any tenable economic theory for its reckless position, the GOP gave up first (just as I predicted).
The moral of the story for Obama: Fighting can actually work -- especially if you are indisputably correct on the issue.