The Treasury banks with commercial banks, but it also has an account at Fed. That said, banks aren't legally obligated to accept its cheques.
> an additional player, the Federal Reserve
That's what this memo describes [1]. In essence, the Fed says it will defer to the Treasury on whether debt is legally issued.
> Treasury sells some bonds to the Fed
The Fed can't do that [2]. What they can do is announce to the market that they won't treat post-ceiling debt differently from pre-ceiling debt. That's what this memo says.
It may not be able to continue doing that if e.g. SCOTUS rules Treasury exceeded its authority. But at that point, the Treasury could start testing its minting power.
In summary, the Fed isn't a player in the legal part of this game.
[1] https://www.federalreserve.gov/monetarypolicy/files/FOMC2011...
I kept thinking about the Fed doing maneuvers as "one possible path", and the Treasury doing maneuvers as "another path", but they're kinda the same.
Thanks.
Why write a check?
So in the scenario you describe, the Treasury writes checks, and they bounce. That would be... bad.
The Treasury would have to collude with the Fed. Something like, the Treasury sells some bonds to the Fed, and the Fed buys them, putting the money into the Fed account. That's illegal, because Congress didn't authorize it, and they try to put the Treasury secretary in jail.
The Supreme Court says "We have no idea what's going on, this is up to you jackasses. Go legislate something." A whole bunch of stuff happens, but it all comes down to DC in flames and the country gives up.