Earlier quoted context omitted.
Saving money is equivalent to investing. Investing is pretty good for the economy.
There is no such relation. You can save cash which does not result in additional investment. When banks refuse to lend money, saving does not result in additional investment, in fact that is the cause of QE. The government feeling compelled to invest because people in the private economy refused to invest.
Stuffing cash under your mattress takes it out of circulation. An inflation targeting central bank will 'print' more money to make up for the missing cash.
That new cash will be spend. (Or, if people keep stuffing it under mattresses, the central bank can just keep 'printing' money.)
In a few years, when you take the money from under your mattress and buy yourself a nice ice cream, the central bank will notice that total spending has gone up, and decrease the amount of money in the economy to keep inflation on target.
It's exactly as if you had lend out the money from under your mattress to the central bank.
(And by 'print money', I mean that they will buy assets with newly created money. Typically they buy government debt, but they can also buy stocks or gold or foreign exchange or rare Lego sets.)
In any case, it looks like we don't actually disagree on what's happening, only on how to interpret it.
I hold that stuffing money under your mattress isn't the problem here. Central bank monopoly on printing money is.
If you stuff privately issued money under your mattress, it's the private issuer that can increase its balance sheet. No government-run central bank required for this.