I'm not exactly sure why the US would default if the debt ceiling were not increased. If the debt ceiling were not increased, the US would not legally be able to borrow but could still pay off interest with revenue from existing taxation. I am assuming of course that the US is not just borrowing money to pay off the interest on money they borrowed earlier.
Regardless, it is clear that the Americans need to both increase taxes and cut spending, especially on useless expenditure. That Republicans are not willing to allow increases in taxes is outrageous. In my opinion, the Democrats should just allow a default to occur and let the American people see just how horrible it will be. Th government can cut spending and not built roads, abandon hospitals, stop paying pensioners, and let banks gouge homeowners and businesses with high interest rates. The Democrats can then blame the Republicans for destroying the economy and in the next election should comfortably win.