bankC creates a secret number, hashes it and sends it to bankA. bankA sends money to bankB locked to hash. bankB can't get money until they have that secret number. bankB sends money to bankC locked to hash. bankC reveals secret number to bankB to unlock that money. bankB does the same with bankA.
Tada, we eliminated the risk of bankB running away with money. This is the lightning network
This isn't a real risk with correspondent banks. Instead, it's counterparty risk: bankB failing while it holds the funds in transfer. That risk can be mitigated with smart contracts, but it's not eliminated. (Correspondent banks also take a portion of the client bank's fraud and AML risk.)
Bank failing in this context would be the bridge (EDIT: contract) gets hacked. Hence mitigated, but not eliminated.
Correct, but in its place is a new systemic risk with a real-world nonzero probability: the contract itself getting hacked. There isn't analogy for this in modern banking since the equivalent issue would either (a) get rolled back or (b) fold into the bank failing envelope. (There is analogy in pre-modern banking, though it largely revolved around debasement and invasion.)
Cast your mind back to 2008 and hopefully this means that one bank falling over doesn't bring down the whole system.