To your question: what it comes down to is that the fed is charged with managing inflation and unemployment. Regardless of the effectiveness of their tools, they are required to do what they are able to do. With that in mind, it’s possible that there is a better course of action.
Personally I find the arguments of MMT to be strong. This forum has charactered the theory as an excuse for profligate spending, however, the actual proponents simply state that budget deficit are ok so long as inflation is under control. Therefore, the theory would posit that the actual resolution to this would be to shrink the deficit spend in order to properly tame inflation.
Take that for what you will.