Dems need their own Ted Cruz. Someone to say: No deal unless it includes a debt ceiling hike and a return to pre-sequester spending.— Joseph Weisenthal (@TheStalwart) October 2, 2013
This idea was simultaneous appealing and repulsive to imagine as an established part of the American legislative system but there is an important reason why the tactics currently being employed by the GOP would not work for Democrats.
The problem is that there is legitimate uncertainty about the willingness to default on the part of a coalition within the GOP. It is generally believed that the majority of the elected officials on the right are not brazen or stupid enough to force a default of the US government. However, the expected probability of such an event is uncomfortably high due to a growing track record of extremist rhetoric and actions. The existence of this asymmetric information is the source of power being exploited by House leaders.
If the current situation were mapped as a game tree, last year's dabble with the debt ceiling set us down a previously ignored path because the expected payoffs to both political parties was assumed to be negative. However, now there is the creeping doubt that some House Republicans mistakenly believe that there is a positive expected payoff waiting for them at the end of the road.
The same strategy would not be credible if threatened by a coalition within the Democratic party for a number of reasons. For one, the Democratic party has exhibited more effective control of its members in recent history and the growth of such a faction within the party structure would likely be powerless. One motivation of rogue GOP members is the idea that they have nothing to lose from staking out this position. Voter groups adversely affected by the tactic of a government shutdown are can ignored for some Republicans. However, Democrats have built their party image around duty and fidelity to the suffering individual which makes a kamikaze threat by the party ineffective.
More importantly, at the core of the Democratic party, there is a feeling of something almost akin to manifest destiny. The history of the 20th and early 21st century has been a steady (and seemingly unstoppable) movement of progressiveness and liberalism. In the last two decades, America has become more liberal economically and socially. Time has been symbolic ally of liberals and conservatives feel the pressure. Ross Douthat wrote an excellent column about why the right is fighting this battle over Obamacare and the debt ceiling that touches upon the pressures of history. The Democratic party doesn't have a feeling of exasperation that has swelled over the previous decades to embolden them to take extreme measures.
Without the pressure of history and endogenous movements within the Democratic party toward a fervent ideology, there is no asymmetry regarding the intentions of Democrats. Such threats would sound like hollow, desperate attempts to fight fire with fire.