Yes. The next recession will just be worse since Fed cannot lower rates any more and if they signal any sign of pulling back on propping up corporate bonds or mortgages those markets will just crash.
The Fed is not focused on the stock market. When they improve the status of the economy through monetary policy, they indirectly improve the value of publically listed companies. This makes sense, because companies are the central entities in the economy. I don't know how the Fed could improve the state of the economy without affecting the prices of shares.
Take a look at December 2018. There was no pandemic. The Fed tried to very slowly reduce its balance sheet. The stock market threw a major tantrum (by dropping 20% or so) and voila, the Fed reversed its course.
The same in 2016, and other times
How can you say they are not focused on the stock market? They are primarily focused on propping up the markets.
If you think they are propping up the stock market, then it doesn't make sense to be worried about inflation. Inflation would devastate the stock market.
They shouldn’t be but this is not true now. What they’ve done has directly propped up asset prices. No question. This isn’t a second order effect. They know what they are doing.