This isn't correct by your own link's information (it says no one owns it). However the Federal Reserve is run by a government appointed group who then oversea the various Federal Reserve Banks, each of which are incorporated and have stock which is what the private banks own.
It does seem a bit convoluted though.
The US Government considers the federal bank “independent within the Government”.
The Fed derives all its authority from the Federal Reserve Act, legislated by Congress, and it’s run by a board appointed by the President and confirmed by the Senate.
It's one of my favorite stocks: I never have to worry if the value is going up or down.
One bit I'd nitpick is restricting the mission to non profit. Profits could still be good, if shareholders have the right to decide what to do with them. Non profit structures force you into spending "above the line" which muddles things by design. Having a profit is a useful health check, and profits can be spent in a more transparent way.
Imagine such a structure owning street level retail real estate, a busy shopping street. Renting at market value to H&M produces more "profit" than renting at reduced prices (or free) to a community op shop. There's a trade-off that the "owners" might make, if they would like to buy more real estate or whatnot.