Leverage

Here's the thing: it's only really "leverage" if the President doesn't want the U.S. to default and the Republicans don't mind if it happens. That difference is what gives one side leverage over the other.

If, on the other hand, both the President and the Republicans don't want default, then neither side has more leverage over the other. In that case it's just a game of chicken, with both sides bluffing and then trying to see who blinks first.

Except that the President is not bluffing. He's saying straight up, that he doesn't want a default, neither do the Republicans and so he won't negotiate over it. The President has caved before, but if he manages to stick to his guns now (like he did on the last debt limit showdown, but not the one before that), then the Republicans will ultimately crumble because they really are bluffing. They don't really want a default. They are just betting that the President will prove to be as weak kneed as he was prior to his reelection.

Which means that really the leverage is all on the President's side. All he has to do to exercise it is to not be Captain Caveman anymore.