Notes are a liability of the Federal Reserve. Meaning they can be traded for balance on a Federal Reserve account. The treasury can't just create liabilities for the Fed out of thin air, I mean that would be wrong ;-)
(edit: just joking. The $1T coin would also be deposited at the Federal Reserve. I don't know why it can't be a note, other than that's how the law was written.)
But when it comes to authority, the constitution requires issuing IOUs that come with the full faith and credit of the US goverment originate in an act of congress. This is because "full and faith and credit" is not something that the executive can do on its own.
The constitution also gives the executive the right to mint coinage, but only up to spending limits, except for a loophole for various collectible coins made out of platinum. These were not intended to circulate, or be deposited at the Federal Reserve in exchange for reserves, which is why I am skeptical about the legality of using this coin. It would also be a huge political embarrassment. But if they cannot deposit it for reserves, then no one is going to get their bill paid with the trillion dollar coin.
In the past, congress had to pass a bill for each debt issuance, which was too much work as debt issuance ramped up. Then congress went to a debt ceiling authorizing the issuing of debt up to a certain amount. This was intended to give the administration flexibility. For the same reason, many spending programs authorize payments "up to" some amount in order to support flexibility. Should the executive do this, you can be sure this flexibility would disappear and things would get much more painful for the executive.
Now if you don't see anything wrong with bypassing congressional authorization because conceptually writing an IOU on metal and paper is the same, then we fundamentally disagree. It's not about the act, but the authorization behind it.
Congress can fix all this with a stroke of a pen, but eliminating the debt ceiling or lowering appropriated spending below what the debt ceiling would require. This is just a cheap trick to allow the executive to bypass the debt ceiling/government shutdown shenanigans.