As I said before, I don't understand your example of BMW lending cars, so let's keep it simple and proceed now from bitcoins (apart from gold). FRB is irrelevant here, and to make things even simpler, let's assume there are no reserve requirements at all (a good assumption since there is no Central Bank to begin with). Could the banks then create bitcoins out of the thin air just like they are said to do with fiat?
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 believe that this is the most likely scenario (second most likely being the demise of Bitcoin). We were here before 100 years ago, so I wonder how it will play out.
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.