Firstly it's going to be hard nut to crack about banks finding themselves embracing the crypto technology, all other means may seems futile at some point cause the banks and crypto community are two different people altogether and hence the banks will always find a way to exercise it's power over crypto but my thoughts on how they can get involved in it fully is simple, there should be an easy adoption first then a medium in making seemless transactions.
You would assume that, but wait until banks start to offer crypto safety like vault, and you will see billions go into it. Like for example blackrock or all these ETF's are not so far off from banks, and yet after they got into bitcoin, billions went into them and that is what we are seeing at the moment. If banks went into it, then we would have seen the same thing again as well, it wouldn't be really that different and billions would go into it again.
The exchange part is difficult, but keeping your bitcoins at some bank wouldn't be really that different or difficult than keeping it at some exchange. Normally people "say" they are against keeping their money at exchanges, but there are over 100 billion dollars worth coins in exchanges, some of that would go to banks.