Earlier quoted context omitted.
You’re describing only one flavor of inflation. What the Fed is supposed to care about is price stability, and lots of things influence prices, and not all of them are money supply and/or new money supply correspondent. We saw this during COVID, the prices of things shot through the roof because of a combination of supply-chain shocks as well as the already well capitalized seizing the opportunity to spend their war…
Inflation isn't like ice cream that comes in assorted flavors. It's a simple ratio (hard to measure in practice, but still conceptually simple). If one side of a ratio is fluctuating (for whatever reason) the ratio's value can be stabilized by making corresponding adjustments to the other side. Amount of goods drops 10%? Reduce the money supply by 10%, bam!, no inflation. There may be all sorts of policy or political…
That's something that's interesting to debate. Especially given recent experience on the opposite side when loosened COVID monetary supply in response to supply limitation boomeranged into inflation by turbocharging demand.
Ultimately though, tighter monetary policy will (full stop, no if's) act as a brake on inflation, explicitly because it will reduce aggregate demand. Less money to chase goods => some demanders stop trying => lower prices.
That 'demanders stopping' is often also an economic slowdown is a different issue.