In a "permissioned ledger" (i.e., a distributed mirrored decentralized tamper-resistant database for a closed set of non-anonymous, legally bound entities), transaction processing would be done by the member entities, for whom the service would be compensation enough; and/or by external contractors, who would get paid in dollars through banks, the old-fashioned way.
They already have what you've described. It is called SWIFT.
Yes, I am sure that pretty good solutions that problem were known and used for many years before bitcoin. Those solutions may not be universally used for many reasons -- including inertia and risk avoidance.
Or safety. Delays of hours or days in interbank transfers are an important safety feature, and maybe exist for that reason alone. When instantaneous transfers are possible, bank hackers and money launderers often take advantage of them, by passing the stolen money through several banks in quick succession, to delay the investigators. Kidnapping and armed robberies also becomes easier and safer, since the ransom can be paid from the victim's bank account and cashed out before the police becomes aware of the crime.