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.
It does not solve the essential problem of there being two claims to the same property simultaneously, which is the fundamental flaw in all Fractional Reserve Systems. I agree that people should be free to bank with whomever they choose, fractional reserve or otherwise, but you should do so knowing fundamentally that the institution is bankrupt.
Fractional reserve banking only works[1] because of the FDIC insurance system. Depositors will be bailed out with depreciated funds due to their bank's profligacy and that's what creates the trust which exists in this system. When the confidence that the FDIC will be able to pay depositors, well, then the system has reached its limit.
Bank Runs in the classic sense don't happen now. Digits are moved from one bankrupt institution to another of slightly higher perceived solvency.
[1] - for very small values of 'works'