But that's not an answer to my question. I'm asking if - regardless of how policy is being executed now - money were distributed throughout the economy, would that accomplish the goals of monetary policy more effectively than the way it is traditionally carried out?

It does answer your question - what you're misundestanding is that "distributing money throughout the economy" is fiscal policy, not monetary policy. That's what the OP meant by "fiscal policy is necessary" and "the government dropped the ball on fiscal policy". The Federal Reserve, as a body, and monetary policy in general by extension, does not posses the tools to enact fiscal policies.

EDIT: oops didn't see OP already clarified.

But monetary policy is also injecting money into the economy. Both fiscal and monetary do this, but differ in two ways: the nature of the money injected and the way the injection is performed.

Monetary policy increases the money supply itself through the exchange of money for treasury bonds and relies on banks to distribute this "new" money, while fiscal policy redistributes money that already exists (previously collected through taxes) and gives it to active businesses through government purchases rather than to bond holding institutions.

The interpretation that banks distribute that money is wrong. Banks don't lend out reserves: https://www.kreditopferhilfe.net/docs/S_and_P__Repeat_After_...

Distributing money to the economy = fiscal policy.