No, I wouldn't say that. The treasury is funding the debt that congress has decided to create. That's there job, they have no choice. However the net result is Janet is in charge, because the fed can't let the govt. not pay their bills, the currency would collapse.
Who is to say they fed doesn't want it anyway. The fed needs to give the illusion of fighting inflation, when they know themselves that their tools to do so are blunt instruments at best, and liquidity is better for the banks anyway. They would much rather have inflation over another Lehman Brothers.
The fed must keep the bond market functioning. Remember through the GFC and the pandemic the largest buyer of treasury debt was the fed itself - most of what's on its bloated balance sheet is govt. bonds.
Powell was first nominated by #45, then again by #46. Presidents nominate fed chairman's they are told to nominate - is my guess. The bankers have always been in control. Remember Janet was fed chairperson before Powell, nominated by #44. Powell was an investment banker before he was on the fed (and is reportedly worth $100M). So its a cozy group to say the least.
Neither wants to be responsible for what happens in an election. They like their cozy little arrangement that survives whomever is in office - so they stay out of the spotlight.
edit - I will add to this. Yes inflation peaked at 9%, but its about 19% total since the money printing started - so about 5% a year, and were now at 3.9%. So all this wailing and nashing of teeth resulted in 1% decrease in inflation. A lot of it was transitory after all.