States will definitely apply blockchain technology in various sectors of the economy and management, since this technology, when using the Internet, makes it possible to securely store any databases in constant access mode. However, when creating their own digitalized central bank currencies, this will not always be the case. Some states claim to be developing their stablecoins on the ethereum platform and will therefore use the blockchain. At the same time, the Chinese government does not use blockchain technology to create its digitized yuan. Let's see how it will work after its official launch.
The banking system will exist as long as the states themselves exist, since they are part of each state. They will apply something new to the extent that it is economically beneficial for them.
Realize that when you are using blockchain the default is p2p, which means that it is a public ledger and nations, specially governments, do not want to make everything a public ledger. There are so many things that they are keeping it a secret, hell in my nation there is even a huge 100+ million dollar "secret fund" that politicians voted for, and interestingly those politicians normally say exact opposite of what each other say and they hate each other and even sometimes even fight each other, they hate each other that much yet when it was about 100 million secret fund they all accepted it.
This is why I honestly believe that they will not use a public ledger like blockchain bitcoin has. Maybe they can find a secret way to do it, I do not know how they can do it, but eventually they may do it and this could profit them a bit, I do not know how much but there is a chance they could develop something like that.