The quote from the Fed site makes for some great reading. Really read it and try to undestand what it is saying.
"it is an independent entity within the government, having both public purposes and private aspects."
What exactly is an 'independent entity'? They say they're not private. But they're not part of the government so they're not public either. They're just 'within' the government. They're not private. They're not public. Is there a third option?
Also, look at the misdirection in the final paragraph you quote. This deals with the 'ownership' issue of the individual banks, which is different than the ownership of the Federal Reserve System itself.
The Federal Reserve Banks of each region are owned by (issue their stock exclusively to) the member banks of that same region. The member banks are privately owned corporations. You can't trade their stock on the stock market, so what? It doesn't stop them from being privately owned. There are lots of companies out there that are privately owned and not publicly listed on the stock exchange. Thus the Federal Reserve Banks are privately owned.
These 12 regional Fed Reserve Banks make up the Fed Reserve System. Do 12 private banks that make up the System result in a System that is 'indepedent within government'?
A quote from this
website about this so-called independence:
"A little independence is a good thing, unless that independence is in reality virtually total and the entity involved controls the nation’s money and economy. Think about it: if the Fed is independent from the government that created it, then who controls it – it has no brain of its own – it is not a person. If it is not controlled by our government, then by whom?"