Economists look at inflation in many, many ways. I don't think anyone that's reasonably well informed, especially economists, misunderstands the cumulative impact of price changes.

Economists that make monetary policy decisions look at recent inflation trends + projected inflation because they are tasked with price stability, which requires them to often respond to shocks well outside their control (war in Ukraine, massive government spending, tax cuts, covid-19 pandemic, etc.)

I was trading and researching fixed income and inflation markets (and implementing in multi-billion dollar portfolios) years ago when you had inflationistas claiming the Fed was going to cause double digit runaway inflation. At the same time, you had people claiming the Fed was not doing enough to support markets.

No matter what monetary policy makers do, it will be pretty much universally mocked by pundits and especially anyone that wants to talk their own book.

Academic economists don't really focus much on any particular reading of inflation, unless perhaps they have their own axe to grind about how it is measured or responded to.

Monetary policy can't change the past, which is why they evaluate current and expected inflation, not what happened in two years ago. Just because prices increased dramatically in 2022 does not mean the Fed or any other central bank should aim for deflation.