“A few points here...the Fed only purchases Treasury securities from Primary Dealers (select large banks). They do not purchase anything directly from the US government. Therefore, any currency used in the purchase goes to banks - not the government.”
The dealer gets a cut, and although the funds may temporarily sit in accounts, the funds are not permanently under the control of the banks. The funds are eventually spent as directed by law by the legislative branch
The Primary Dealers by US securities outright during Treasury auctions. The Federal government receives currency from these auctions to use for its purposes. The Primary Dealers are then free to hold the securities and do whatever they want with them.
At certain periods, the Fed will purchase some of these securities from the Primary Dealers in order to 1) circulate more physical currency into the market - since banks need new dollar bills for their customers, and 2) to unethically prop up the big banks.
In case #2, the currency that the Primary Dealers are paid by the Fed mainly stays sitting in the Dealers accounts at the Fed. Look at the weekly H.4.1 Fed report - Section 8. It shows that there are over $2.14 trillion in deposits just sitting there - collecting 3% interest.
There is no governmental law telling the Primary Dealers how they must use their securities and interact with the Fed after an auction has taken place.