The power to borrow is to write a promissory note in exchange for a real something, that is later redeemable by giving real somethings to the holder of the note. I can write a note promising to pay a hamburger and a half on Tuesday for a hamburger today. In order for that note to have any sort of economic effect, someone has to give me a hamburger for it.
Once it's out there, that's worth 1.5 burgers, and can be traded around at will, but come Tuesday, I need to have 1.5 hamburgers on hand to redeem the note.
The paper money notes now circulating in the economy have nothing to do with borrowing. Where is the promise to pay? What is the term of the loan? Why is the US paying the Fed interest to take and use their note?
If it were borrowing, the Treasury would print the notes and give them to the Fed in exchange for something. Those are T-bills. The something they get in return is Fed scrip. That is what circulates. That is what they made into legal tender. Do you buy groceries with T-bills? US notes? Gold or silver certificates? No? The federal power to borrow does not apply here. You are not exchanging federal instruments, but notes issued by a private (but legislatively controlled) banking system.
If you contract to be paid only in federal coins or instruments, and someone pays you in private bank notes created from thin air instead, the courts will not decide in your favor.
The power to emit debt instruments, including in the form of paper money, is, to the extent it is not included in the power to coin money and regulate its value (Art. I, Sec. 8, cl. 5) a necessary part of the power to borrow (Art. I, Sec. 8, cl. 3). See, inter alia, Juilliard v. Greenman, 110 U.S. 421.