I believe that if Bitcoin becomes mainstream, a fractional reserve banking system will develop and there will probably be one or more central banks. These central banks will be created by banks or their regulators and may or may not be government-controlled. They will pool member deposits and provide deposit insurance in order to reduce the impact of bank runs and they will have to set reserve requirements.
I doubt that. Not that someone won't try FRB with Bitcoin that is almost an inevitability; bankers are going to banker. Central banks would not have much interest in a system where they can't "magic up" more money in the monetary base. Also without the unlimited printing of central banks we would have seen the complete collapse of the banking system worldwide and probably the end of one or more modern currencies. You can't do that with Bitcoin. It is more like FRB using gold deposits. Central banks are a lot less important in a system where they can't pretend that they can shape and control the economy.
It will be more like the original central banks when there was a gold standard. The banks' motivation for forming central banks will be the desire to get customers to deposit bitcoins with them. Without protections such as deposit insurance and a lender of last resort, it would not be wise to trust a bank with your money. Governments will probably get involved because it is in their natures.
Why would you want to deposit your bitcoins into a bank? Bitcoin value can grow by itself even with no help from the bank with its interest paid on deposits.
Why would you turn down risk-free interest on your bitcoins (if it were available)?