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.