you are exactly right, but everyone will be using it, as all fiat will go across this blockchain
I doubt it.
If you operate a backed currency you're far more efficient using a secure SQL server as the banks will soon find out to their cost once the "we're blockchain friendly" bandwagon gimmickery has worn off.
The whole point of a blockchain is that it's designed to support peer-to-peer, unbacked tokens. Ergo, it's not much use for transporting non peer-to-peer, backed ones. Even if it does shave something off their clearing expenses it isn't going to change the nature of their money which will still require willing debtors to queue up and underwrite it with their future labour. (And those are in increasingly short supply).
If they have only a few private bankchain "nodes" doing the "mining" then you only need to hack one and use an ASIC miner to 51% attack the network and increase the difficulty to the point that when you pull your ASIC off of the network they either have to upgrade to ASICs to keep going or the difficulty will not allow any further blocks. Or use your 51% to doublespend.
