IMO, the risk to the economy isn't what happens when the $200 billion in buys take place, but when it stops. Take the QE of the 2010's that caused interest rates to artificially decline - that pulled forward demand, but also created sellers, as they were more inclined to move to a nicer house while interest rates were favorable. When the Fed became boxed in after COVID and interest rates reverted to market levels, the only sellers were the estates or those moving to nursing homes. The increasing population pushed an imbalance between supply and demand, so those that would have been candidates to buy a nicer home elected to stay with their current home, rather than pay significantly higher prices and at much higher interest rates.
The first-time homebuyer tax credit is another good example - shortly after it ends, there was a complete absence of activity in the lower-priced neighborhoods, outside of short sales and foreclosures that were snatched up at fire sale prices.