The answer upon the question "Why should do banking on Blockchain" is already hugely discussed. I am more interested on the question "How to make cross-platform transactions easily while still ensuring the privacy and security".
They do not want that, they want you to only use their systems so they can have complete control over your money and your life, so while I can see why people want that since it will be convenient for them, banks do not want that and we should not expect a huge leap forward, so we need to keep pushing for the direct adoption of bitcoin and not worry too much about what banks do.