I take a very different view than you do over the Fed. For example, look at the very slow economic growth following the great recession yet the stock market boomed. That was totally driven by the Fed. You remember the 2013 'temper tantrum' the market throw when Bernanke said he was going to take the punch bowl away. The Fed said they were pushing for the wealth effect by basically creating assets bubbles in equities and housing. The Fed's role in stock market started well before TGR of course but their actions affect the market more than any President does.
First, I agree completely that the Fed has far more to say about the economy than any President. As for QE, the Fed did not and should not concern themselves with the imnpact their decisions have on the stock market. The Fed bought a bunch of long term government debt which helped restart the housing market and they bought a lot of the bad debt from banks, but recovered almost all of that. It's natural that those purchases drove down interest rates, especially long term, and caused a spike in the market, but that should NEVER be a factor in the Feds decision making.