Right now they are all on TV using the ‘tell the GOP not to default on our debt’ lines to get as much milage out of that as they can.
Technically they have a partial point, the House while they sometimes feel its at gunpoint, passes tax and spending bills that creates or at least adds to the deficit, but they also inherit tax and spending bills from past congresses which they cannot repeal by themselves that also adds to the deficit.
In the end like the other fights this will like the others end based on which side most voters support, even if that is not a reality many here wont accept,