Fred Bauer warns Republicans that the new method they've pioneered -- using the debt ceiling as a hostage to force policy concessions -- could just as easily be turned against them under a Democratic House and a Republican president:

In passing the Ryan budget, Congress also upped the debt ceiling by a trillion or so, but perpetual deficits mean that the ceiling is coming awfully close, and federal spending is due to break it in early August 2015. So now, in May, the president must go on bended knee to Speaker Pelosi, who demands tax increases as the price for her caucus supporting an increase in the debt ceiling. ...
And what could Republicans say to this?
I appreciate the spirit of the thought experiment. And Bauer is certainly right that Republicans would have no legitimate procedural argument against this tactic, though you can always whip one up if needed.

But is this really a plausible scenario? I don't actually think it is. I could imagine a Democratic Party holding the debt ceiling hostage, but not this Democratic party. It would have to be a far more left-wing party, in which activists have gained greater control and which has largely severed itself from any business influence. The current democratic party lacks anything like the will to power to threaten economic catastrophe in order to force a government mostly controlled by elected members of the opposition to accept its contested policy agenda. And it would require a substantial coterie susceptible to the argument of the default denialists -- a natural fit for the party of supply-siders and climate change deniers, but not a good fit for the moderate coalition that forms the current Democratic party.

James Surowiecki argues that the debt ceiling specifically favors a certain kind of politician:

[B]y turning dealmaking into a game of chicken, the debt ceiling favors fanaticism. As the economist Thomas Schelling showed many years ago, “It does not always help to be, or to be believed to be, fully rational, coolheaded, and in control of oneself” when it comes to brinksmanship. It doesn’t, in short, help to be President Obama. That may be why all the deals that have been taken seriously this season rely much more heavily on spending cuts than on tax increases: the deals represent Republican priorities, because the Republicans seem to be more willing than the Democrats to let the country default.

There's only one party in American politics today willing to try this sort of thing. If history had run a different course, and the Democratic Party had been captured by the ideological descendants of Tom Hayden, and the GOP by the descendants of Nelson Rockefeller, the situation would probably be reversed.