Right, but the US has enough income to pay the debts it has already incurred, which the president is constitutionally required to pay.
Right, he won't be able to borrow more, but he will be able, and he will be required to, service the debt that has already been entered into.
OF COURSE the president controls the purse strings.
The Treasury, the purse, is part of the president's executive branch. To say the president doesn't control the purse strings is to miss the fundamental design of the US government, and the separation of powers between the branches.
Yes, Congress has to authorize access to the purse, but once that authority is given, the control is in the hand of the president, as he is head of the executive branch.
But the US doesn't need to borrow more money to service its debt. The Treasury will bring in enough money in tax revenues to service the outstanding debt. That's why all of this talk of default is mathematically silly. The Treasury has more than enough money to avoid default regardless of the debt ceiling.
Legally the president cannot order default, but technically it's up to him. If the president chooses not to service the debt, that would be an impeachable offense in my opinion.
But because the president controls those purse strings we need to hold him accountable for it, and call him out for it.