Go look at:
https://www.federalreserve.gov/default.htmOf course the Fed is an arm of the government. The member banks have the "form" of private banks, but they are not really private banks. They are government created, backed, run, and funded entities. Think of a Port Authority, and decide whether it is part of the government or not. Of course it is.
The member banks also have no power in Fed decision making, which rests solely in the hands of the Chairman. The members banks exist solely to provide banking services for commercial banks in their district, and as various jobs programs for economists. Some also have museums and exhibitions.
Back in the day when you travelled the nation on horse and buggy, if there was a banking crisis it was important that there be a nearby bank with money. So yes, you needed a network of local reserve banks. Today, that's not so important anymore.
The distinctions being made here are due to a (quaint) stock issuance system that maybe made political sense in the politics of 1913 - when politicians still remembered Andrew Jackson and the Bank War (https://www.history.com/topics/19th-century/bank-war) - but this symbolism is irrelevant today.
Don't get me wrong, there are lots of legitimate questions to ask about the Fed in the zone of public/private -- is the Fed acting in the interests of the financial sector or the interests of the public? - how "independent" is the Fed"? But arguments about the Fed being really "private" are deep in the realm of conspiracy theory pamphlets.
Anyone who tries to argue this immediately loses the argument, in the same way that people arguing about the influence of the Rockefellers immediately loses the argument. Maybe in 1913, these were useful questions, but today these are all cringe questions.