One day they will accept. It's just a matter of time now. The first steps towards adoption they have already done. For an example, banks in my country offer Bitcoin as investment alternative, although you can't access your Bitcoins directly, without using the bank. However, it doesn't have to be like this forever. Soon I believe they will become less strict, allowing customers to deposit and cashout their BTCs to external addresses, although some kind of KYC must be implied along the process.
The question is: Will regulators allow it? The US is a pro-crypto country, so it wouldn't have a problem with this. But what about other countries such as those within the EU bloc and BRICS? I'm pretty sure they won't allow banks to offer crypto services to their customers. Besides, Bitcoin is NOT legal tender. So banks would be "playing" at their own risk.
Why care about banks, though? Isn't Bitcoin good on its own? With Bitcoin, you can truly be your own bank. I guess the OP doesn't understand what Bitcoin is all about. It's more than just money or "digital gold". It's a movement, a revolution meant to bring back monetary freedom to the people. I hope it fulfills its purpose, before it's too late.
Actually base on this article there are banks already accepting it
https://en.zcredito.com/banks-that-accept-cryptocurrency-deposits/So there's really good chance that more banks would follow especially that there are few banks initiate this acceptance already.
But for me I don't really care if they accept it or not since I want Bitcoin to be in my full custody. I don't want bank to hold my funds then get profits with it.
I wish that day never comes. Instead, I expect a day when bitcoin has completely replaced banks.
But unfortunately that situation you are expecting or hoping for won't gonna happen since for sure banks and fiat are first priority of government so for sure Bitcoin will be just be their option.