You're missing that the Constitution also assigns authority to borrow against the credit of the United States to Congress, which is really all the debt limit is, the amount that Congress has so far authorized.
And this is a pretty important authority! Congress should be involved in the issuance of debt since it's about obligating generations of Americans to repaying it. We should at least have the democratic process affirm that the country really does want to take on that debt.
So no, the debt limit is not a manufactured idea that has created an unnecessary stalemate. It is a democratic idea built into the core of the country for really good reasons, and the president needs to get on with paying debts out of the Treasury instead of fighting for more borrowing power.
The debt limit didn't create this stalemate. Political grandstanding did.