Everything is possible, and I believe a time will come when the bank will be the biggest and safest exchange because Bitcoin is still growing. We can see how many countries are accepting Bitcoin and also developing a Bitcoin reserve.
If the governments are now friendly with Bitcoin, then it is possible that one day the bank will have something to do with Bitcoin, and it will be a great advantage to the bank because the bank will be more reliable than online exchanges. There won't be cases of theft and scams like what is found on online exchanges. I'm very sure that in a short time, we can have this.
If banks start acting as an exchange, it will be bad business for the current exchange market. Since leading exchanges ask for KYC, people might prefer to use banks since both of them are centralized. Banks are registered in our country and are also insured. Funds will be safer in banks compared to centralized exchanges that are registered in countries like the Cayman Islands.
The government might authorise banks to act as crypto exchanges, but it will be under strict supervision. The KYC requirements might be very annoying.