Sending bitcoin to the bank defeats the purpose for which it was created. Bitcoin signifies "freedom of the unbanked," so what is the point of sending it to the bank then? Besides, the bank is a third party, and Bitcoin came as a rescue from third parties, and this also includes the banks which uses every opportunity it has to exploit people from their hard earned resources because of the fact that the government backs them up for their dirty acts. Even if the banks accept bitcoin, it will be the last thing I will ever do to deposit my bitcoin with the bank when I can keep it under my own custody.
True. But if banks start accepting BTC deposits, most people won't hesitate to entrust their coins in them. They prefer convenience above all else. This will ultimately enrich banks, as they profit from customers' holdings. Although, I don't think most banks would be willing to take that risk. Especially when Bitcoin is volatile.
Besides, adopting Bitcoin will uncover banks' dirty schemes. It is a transparent ledger, after all. Banks are better off with Fiat because they can manipulate it to "their hearts' content". But who knows? With "Wall Street" in the game, anything's possible. Let's us hope Bitcoin doesn't turn centralized in the long run.
I doubt that would happen someday, just like you said the banks are actually the government or even private users ways of extorting or self beneficial way of milking their customer and i think the original idea of bitcoin itself was to contradict the traditional third party involvement of an unknown entity in financial transactions since whenever you want to confirm a payment or even do one the system of banks allows a third person not involved in the transaction to take full and control of how and when your money get to the receiver ends.