It can indeed! If your bank wants to upgrade their infrastructure to integrate blockchain - and you continue to use your banks services - then you indirectly add to the success of blockchain. Bitcoin continues to enjoy the tag of blockchains only successful application, but the initial outcomes of latest projects have proven that blockchains can be integrated as private ledgers to improve companys record keeping and transaction speeds.
However, the private blockchains will be nothing but a close equivalent to private databases. Bitcoin, on the other hand, will always enjoy the perks of being decentralised. Im sure that people would want to stay attached to an independent economy.
Blockchain can survive without Bitcoin; it cannot beat it.