the Fed has a dual mandate, there are still a lot of people who doesn't have a job due to the pandemic.

this inflation spike due to the combined effects of overspending on products in quarantine (partly fueled by unemployment checks, that should have been sent in monthly installments to discourage spending it on big items), and the energy market chaos thanks to the war in Europe. (and even though a small portion of the supply disappeared the price curve is steep, the new price point is much higher up as we see.)

should have the Fed done more about this? yes, definitely. but just as you observed, the low rates are here to stay on the long run. (mostly because aging population of the developed countries as pension funds buy bonds to get the fixed cashflow they need to pay the pensions.)