I personally think we trade too much on the side of reducing inflation, and not enough on the side of countering unemployment.
And then there’s a mystery to explain . . .
U.S. Economy over time
https://fred.stlouisfed.org/series/A191RO1Q156NBEA
Gray lines are periods of recession.
Why was the relatively brief 1980 recession followed by the much more serious and longer-lasting 1982 recession?
Because the Federal Reserve initiated a massive rate hike in 1981 to counter inflation, which slowed down economic activity, particularly in areas of the economy where private investment is needed to stimulate demand, like with construction and industrial sectors. The recession in 1980 was arguably ringing in the till for the half decade of stagflation that preceded it. Of course, there is little to indicate that the economy in the 1979 was doing all that much better, but it was not a formal recession yet.
Your point about the weight put upon each mandate of the Fed isn't a new one; I would agree, if we are talking about the last decade in particular, when inflation in general has been hard to generate. This is a facet of having an aging population in a developed economy; Japan is the prototype for this, with Germany and Scandinavia not far behind. They have to resort to negative interest rates (as in, your money has more value in a mattress), to generate investment that could get any kind of normal and necessary inflationary behavior off of the ground. I would say, however, that in the 1970s-1980s, inflation really was the bigger problem and that a sustained attack on it was worth a recession, because serious currency inflation had been chopping at people's living standards for over a decade without the accompanying economic growth needed to offset it.
However, I'd say that recently, we kind of see the limits that monetary policy can have. After all, U3 Unemployment in the US has been sharply diving for a few years now and is reaching some very encouraging levels, even though the Fed has initiated some modest rate hikes. Even U6 Unemployment, arguably a better statistic, has been diving sharply as well (this takes into account those unattached to the labor force). The US misery index is about as low as it has ever been. I'd argue that by the mandates of the Fed, to promote employment and the currency's purchasing power, it has been quite successful. But this doesn't take into account very minute and very selected sectors of the US economy where people are very sensitive to prices, and which monetary policy really has little impact on (like prescription drug prices or private higher education).