The Fed was founded after a spat of banking crises.
We know that the Federal Reserve also has the goals of full employment and steady, moderate inflation. Since the 1990s, that’s meant 2%. But it’s a relatively recent addition to the Fed’s policy goals. The primary purpose was initially and always has been financial system stability.
In 2008, the Fed demonstrated that it’s willing to attain financial stability at the cost of employment. After and during the financial crisis, the Fed purchased mortgage backed securities (MBS) from private banks at a time when their value was highly uncertain (and discounted). The purpose was to replace these assets of uncertain value with less risky assets. At the time, there was resentment that these security holders were insulated from losses while the homeowners whose loans composed the MBS did not get comparable relief. I remember arguing that the Fed, with the cooperation of congress, could have just paid part of the mortgages on behalf of the homeowners such that there were fewer foreclosures and fewer personal bankruptcies. That way, both the borrowers wouldn’t default and the debt holders would enjoy stable returns.
But, the primary goal of the Fed is financial system stability. Pre-financial crisis, banks had loaded-up on securities of uncertain value with the help of regulatory arbitrage and some lending shenanigans. The Fed needed to avoid the ensuing catastrophe that was a consequence of the greater-than-anticipated realized risk. Importantly, catastrophe to the Fed is financial-sector specific. Markets losing liquidity, bank-runs, and financial sector business failures all qualify as the stuff of concern (all of which occurred). While making mortgage payments for specific mortgages would have been popular amongst many debtors, it also would have taken much more time to implement. The Fed wanted to avoid more financial instability than had already occurred. And frankly, the Fed’s first priority isn’t to take care of the public. Given the alternative between a slow popular option and a quick adequate option, the Fed has demonstrated an inclination toward the latter.