Blockchains are just databases rebranded to be something interesting when cryptography is applied. There's no magic to keeping transactions in 'folders'. Database journaling, and message signing has been performed since the 80's (or older)
a database contains data that can be deleted and/or overwritten. Other than examining changelogs and reverting changes according to 'external requirements', a database is a non-auditable pot of changeable information. A blockchain cannot be re-written without a "hard fork" of the network that makes it exist. It is that simple. Consequently,you can also easily audit a blockchain or any part (wallet) of it. To audit a database is not impossible, but auditing a ledger is entirely simpler.
Banking people throw "blockchain" around as a buzzword without necessarily understanding the total difference in the way they operate compared to databases. The fact that the blockchain is a ledger is best demonstrated to them by the notion that a printed bank statement is a 'de facto' ledger of the transactions on a single account. Banks have always told customers that keeping printed (usually monthly) statements is how a permanent record is kept of your transactions. This is because they routinely wipe system transactions to avoid the storage and security costs of keeping a copy of the data. I have bank statements on paper going back to my very first bank account and transaction. They take up a great deal of physical space, but they do represent a permanent and uncontestable history of my finances. They are my personal paperblockchain. If only banks had been willing to keep everyone's transaction data in perpetuity then I wouldn't have boxes of paper in my attic. But that would be hugely impractical and costly. Blockchains for every account are feasible, but not on the horizon given the state of current cryptography and the huge data costs.
Banks trying to get into blockchain tech are only looking at high level intra-bank functionality, which would be well served by a new system, especially given that they don't trust each other but have historically agreed that they should, and pragmatism has cemented the system. A new system is absolutely needed.
rebranding is hardly the point. Blockchains exist as something entirely novel and challenging to legacy database structures. The folders you refer to - do you mean cryptocurrency wallets? The wallets where people can be their own bank? They are just blockchain 1.0 btw.