The problem with that argument is that the Constitution itself assigns to Congress authority over borrowing, which is probably a pretty good idea since you want to have national buy-in before obligating generations of citizens to paying back debts.
So not only is the debt ceiling constitutional, it's actually constitutionally mandated.
The president has to pay the debt. He really needs to stop threatening not to do so.
All of this drama and sensationalist political grandstanding is a distraction from the simple fact that the president has to order the Treasury to pay those debts.