No, but in order to get access to the Fed, the bank link systems and so on and so forth, they need to get government approval.
In order to be bailed out, which is a necessity, they even need contacts as high up as the president.
And even that is ignoring the long history of the government making specific laws specifically to benefit or destroy a single bank.
Nope, no government role here at all. This division simply results in privatizing the profits, nationalizing the losses in trade for having politicians name the owners.
it is if you want to be a reserve bank of course, which is why anyone wants to have a bank in the first place.
Not true. Literally by design actually. This is a common misconception about Central Banks and it's one the US system makes worse by being explicitly public but functioning extraordinarily similarly to a private company.
This particular structure was actually a compromise between various other possible implementations.
But seriously, it's not a private bank. Repeating that just makes you looking like you get your information from websites that put their titles in all-caps and have dodgy grammar.