Yes, that's how the system works.
Where they are being foolish is using this new money to buy federal debt. That's not the smartest way to use the money; that debt will have to be serviced by the government forever, and it creates a net drag on the economy.
While there are competing ideas as to the most effective use of such new, fiat Fed issue, I think that using it for straight up consumption is probably the most effective way to use it although the hardest to defend in our current system.
I personally think that the government might as well simply have the Treasury create the issue ("Greenbacks", not Federal Reserve notes) and use that new money to directly meet expenses. The advantage is that that money goes straight into the economy instead of building yet a higher mountain of debt for our children to pay for.
The downside of such "hot" issue, though, is that it is extraordinarily inflationary. It is the very definition of "running the printing presses".
If we were to do it to the extent of $600 billion of new issue with no corresponding debt against that issue, it would be a very dangerous act since the multiplier on that is probably close to 7 giving (again, very, very roughly) a $4 trillion kick to a $14 trillion economy which would result in perhaps 20% inflation. Or maybe a lot more, or (conceivably) a lot less. It depends on how sluggish the economy stayed with such a powerful infusion of hot money.
‘which would result in perhaps 20% inflation’
Bill Gross, is that you?