Article 1 of the Constitution gives only Congress the power "[t]o borrow Money on the credit of the United States," and the debt ceiling is what constitutes authorization for the Treasury to do so. The President has no authority to override the debt ceiling, or maybe a better way of putting it is that the Executive Branch has no authority or authorization to issue debt in excess of the debt ceiling. The "cleanest" way to get around the Congress on this one would be the platinum coin, and I'd argue that creating a constitutional crisis is the dirtiest way to do it. If the President won't do the former, he certainly won't do the latter.
Yes, but section 4 of the 14th amendment say "The validity of the public debt of the United States, authorized by law, including debts incurred for payment of pensions and bounties for services in suppressing insurrection or rebellion, shall not be questioned." The money has already been borrowed, and the ceiling needs to be raised to service the debt. The reason section 4 was put into the 14th amendment was to ensure that Southern lawmakers who were returning to Congress wouldn't try some sabotage tactics to hurt the country. What these modern Congresspeople are doing is certainly nothing but sabotage. But, just today the President reaffirmed that he will not invoke the 14th amendment. He wants the game of chicken to continue, apparently.