I measure government efficiency by looking at the value it receives for the money it spends compared to a private company. You are focused on government's role with respect to the monetary system.

At one level, government operates at the local, state, and even federal level as the sum of individual cost centers each empowered to spend a designated number of dollars based on their respectively approved budgets in exactly the same way as a corporate cost center. I have run a large company and been a government health and defense contractor CFO. They operate the same way. I have also been involved with working with government health care agencies to "budget" their expenditures. The one thing different about a government agency that I have not seen in the private sector is the decision to stop spending when the budget has been expended even when claims against the government exist. Those claims get put off until the next fiscal year or funded through supplementary budget approvals. Private enterprises operate a bit differently as not paying one's bills on a timely basis carries major consequences.

So, it is easy to measure how cost efficient a cost center is based on the activities it conducts and the cost of those activities. To the extent each cost center gets more services at lower cost, the entity is efficient, or it isn't. I think you are trying to argue the value of fiscal and monetary stimulus rather than simply government efficiency.

That is a different issue.

Regarding QE and money supply, you seem to disagree with virtually every economist.

"Quantitative easing is a form of monetary policy used by central banks to increase the domestic money supply and spur economic activity."

https://www.investopedia.com/terms/q/quantitative-easing.asp