The problem with the fractional reserve banking is that you have the right to withdraw your money at any time. We know that we can't do that all at the same time, and we know that we take a risk of a bank run. But noone forces you to keep your money in an interest earing account. You can put it in a safe deposit box, pay the annual fee, and you can feel secure about your holdings. Even better, you should be able to hold an account in a Federal Reserve bank, and if you need cash, you just go there and cash your federal reserve deposit. And if you are worried about the assets of the federal reserve, you can buy some physical gold.
If there is any problem in the system, it is not in the fractional reserve banking, nor in the fiat currency regime, not even in the Central bank. It is in the power and responsibility we put in the hands of the government. If it is too big and too inefficient and irresponsible, and we do nothing to change it, then any kind of monetary or fiscal system under such government will create unfair distribution of wealth.