Fractional reserve banking is unstable without a monopoly over the ability to print currency, due to the >0 possibility that a bank runs out of reserves.
That's like saying: "Gold atoms are unstable because quantum mechanics tells us that there is a >0 possibility that they will spontaneously decay."
Yes, in the very very very long run the universe will suffer heat death and there will be no gold (or any other atoms) left.
If the probability of bank runs is very small, then fractional reserve banking works. Or, in other words, if banks can establish and maintain trust in their ability to repay deposits they'll be stable even if they practice fractional reserve banking.
I hope we'll eventually find out the hard way if, or which, Bitcoin banks can establish and maintain trust.