To be honest, if banks were the custodian of my cryptocurrencies, I wouldn't mind because the chances of my crypto being lost is going to go down drastically and if it was in an unlikely event got lost, the banks will have sort of insurance that will be in place to take care of the losses.
Banks cannot even take care of the normal day to day probelms, the inflation and deflation it's something none of them can control and at the end many of the banks close down owned by private businesses and the one's owned by the government are under huge debts.
I do feel like banks and cryptocurrencies should be seperate but for sure interconnection is healthy as long as they do not overshadow each other and gets mixed up.
At the same time I have seen banks trying hard to make something like cryptocurrencies but integrated into their system if you do believe you want the support of the banks then go for that. When it comes it Bitcoins, I would prefer it to be Volatile and diverse. Volatility is not bad always. It does have a lot of benefits.