Wrong. The Fed has a dual mandate, they have to keep inflation and unemployment stable. Recessions are bad for employment, it turns out.

You could have a stock market crash while hitting full employment, for any number of reasons e.g. non-listed companies expand while listed contract; families scale down and take earners off the market; war mobilization that involves long-term high taxes; and probably all kinds of scenarios I'm not thinking of. "Stopping recessions" is not an automatic justification for propping up asset prices.

So is the Fed's job to make sure everyone is employed? Unemployment is at an all time low. With a normal spike an unemployment from a normal recession, we would be at normal levels of unemployment.

The fed's target is 2% inflation, not stability of the inflation rate. You could say the relative stability of the value of money is a goal

I think you have it backwards. Stability is the overall target, and the 2% target was chosen as a way to achieve that. See for example https://www.federalreserve.gov/faqs/money_12848.htm.

> The objectives as mandated by the Congress in the Federal Reserve Act are promoting (1) maximum employment, which means all Americans that want to work are gainfully employed, and (2) stable prices for the goods and services we all purchase.

> Following its meeting in January 2012, the FOMC issued a statement regarding its longer-run goals and monetary policy strategy. The FOMC noted in its statement that the Committee judges that inflation at the rate of 2 percent (as measured by the annual change in the price index for personal consumption expenditures, or PCE) is most consistent over the longer run with the Federal Reserve's statutory mandate.