The PoS option was to highlight that power consumption doesn't have to be an issue. Of course, PoS has its own issues.
The network can use any other type of proof, like Proof of Authority where only a buch of validators owned by the banking system can process the transactions. The network can be even tokenless, no profit or incentives from it, just the secure architecture.
Yes, but if we are talking about a private permissioned blockchain, there's no need to obtain returns from staking. It can be even a Proof of Authority tokenless network for what banking care.
Banks are already paying for servers to process and store information. A few validators or collators (quite cheap for a private network) provided by several banks would cost a fraction of what they pay now and they'll keep owning the data, they could reverse transactions, be covered by several layers of public encryption, guard the user's wallet/login, etc.
Don't mix blockchain with the speculative world built on top of it. That's only an unfortunate use of the technology.
All your points are about an obsolete idea of Bitcoin, a PoW public blockchain. A PoS private blockchain with private keys not handled by the users would invalidate your entire list.
The keyword is "private." The redundant system all the banks maintain can be reduced to a private, permissioned blockchain, creating a network for the banking system to handle their own transactions in addition to a seamless inter-bank communication.
I doubt a network for just one bank can be that useful compared to the current situation.
Also, I'd say that every bank has (had?) a team researching the blockchain.