What two competing systems? FRB will exist regardless of whether the asset is $, BTC, or metal.>
Nope, the FRB could survive a longer run only if there is an entity of last-resort-lender, which has an ability to create new units of money out of nothing (i.e. in current fiat money systems - those are the central banks - which uses that to do the bailouts of otherwise failed institutions on regular basis).
With bitcoin no such entity like central bank could possible exist. There is no way to create new bitcoins on demand out of nothing, that's the beauty of it.
Just spit-balling here... Banks could cooperate. Let's say there's a run on one bad bank and its reserve drops to zero. Instead of reaching out to central bank to print more money, the bad bank could reach out to other banks to secure a series of small loans to pay everyone back. The failing bank would need a way to pay it's loans back...so I'm at a loss there. Also, it does become circular at some point, but given a large enough network of banks, one or two failing banks won't bring down the whole financial system.
What if a bank in this sense wasn't thought of as a safe haven, but as a loan co-op? Everyone shares fully in loan decisions, profit, and risk. If a loan goes bad, well, that money disappears. Runs would still have to be mitigated somehow.