The Constitution assigns to the Congress the power to “borrow money on the credit of the United States.” (Art 1, Sec. 8, second clause.) That’s right after the power to tax.
The President can no more borrow money on his own authority than he can impose taxes on his own authority. Any attempt to do so would be a clear violation of the Constitution.
It is not the case that the debt ceiling is a statutory control imposed on an otherwise empowered President. The Congress used to authorize debt issuance on a case-by-case basis, the way it commissions officers. Then it switched to authorizing debt issuance up to a fixed limit. But without Congressional authorization the Constitution does not allow the President to borrow money. Period.
The 14th Amendment says that the validity of the debt shall not be
questioned. It does not give the President any authority to do anything whatever. If the Republicans allow the government to run out of money, the Treasury has to stop writing checks.
Yes, that’s a bad outcome, which is why we should do everything we can to put pressure on Congressional Republicans to do the right thing. Inventing imaginary ways out of the problem just tells the infants then can keep on making messes because the grown-ups are available to clean up after them.
Update Yes, some of the lunatics have gamed this out. Sarah Palin says that it would be an impeachable offense either for the President to not pay the debt or to borrow to pay it past the limit set by Congress.