Why would you want a "government bank" to hold your money?
Most of the bank in America is owned by private shareholders. They are NOT "government" bank.
Ok let me rephrase. Government controlled banks.
Never probably, it's no in their interest, maintaining the actual status quo. You would kill yourself one day if the life of your human fellows depended on it ? Please note the government we have in our times works more like a machine, based on rationale not feelings.