Or they could just maintain their own ledgers for 1 billionth the computational cost.

Bitcoin by itself doesn't actually need much energy - it could easy run in a single server. The reason it uses so much now is because so many miners have joined, so the software increased the computational difficulty to keep generating 1 block every 10m.

If you run the bitcoin daemon in "regression test" mode, which creates a private, disconnected blockchain, it can actually generate ~200 blocks per minute using just a single CPU.

If a block chain is closed and the membership known, you could use some metric of "organizationally distinct signatories" to determine the length (/cost) of a chain rather than proof of work.

Don't they do that already? What is a 'closed blockchain' if not a sql server? The only value a blockchain has is as a public resource.

A closed blockchain is still distributed (just not open to everyone), and still has different nodes checking up on each other, while an SQL server is a single point of potential fraud.

Do banks process actual transactions through a SQL Server?

Sure, where do you think Oracle gets all their revenue? :) I mean, interbank transactions go through a clearing house like FedACH, but even that is just a remote RDBMS.