The Fed being private is a legal fiction to do an end-run around the Constitution. It exercises powers that the government could not legally give itself, so the government instead created a third-party, claiming it's not part of of the government, and gave the powers to them instead.
Now, if you read this and said, "that's stupid, they can't give powers to someone else that they don't even have," well, I agree with you. Too bad they managed to get this one past the courts, but they did manage to do that, and now it's been over 100 years, so it's pretty firmly established.
In practice they are part of the government. They act just like they're part of the government. They work hand-in-hand with the government, they receive their appointments from the President, and Bernanke even has said that he viewed himself as "part of the administration" during his tenure at the Fed.
We would in no way be better off to have the Fed be officially run by the government, than we are with it private. It would be exactly the same.
The problem isn't whether it's private or not, the problem is the Fed's existence. They have brought us the devaluation of our money by 98-99% of its value, the Great Depression, the stagflation of the 70's, the dot com bubble, the housing bubble and crash in 2008, and the current bubble economy that is just in the process of going down the tubes.
They suck at their job because everything they think they know about monetary policy is wrong. We'd be better off with no central bank at all, with gold and silver still being the official currency of the land. Or with various private currencies, including cryptos, as well. We'd be vastly more prosperous.
A free market in money is what this country needs. We learned through the Soviet Union's failure of the disastrous nature of central planning to a nation's economy. Why, then, do we think we can succeed by centrally planning the one single price that involves the entire economy--the price of money--when we know that central planning doesn't work and leads to ruin?