Assume there's a legitimate company out there that will verify driver's licenses. Do they really need to store everything "online"? No, they do not.
They can store it "offline" with a 1- or 2- minute delay to access the data, caching it in "online" systems for maybe 15 minutes.
This, plus alarms when large amounts of data is being moved from "offline" to "online" in a short period of time, would've either made this leak slower than it was or, more likely, led to this year-long leak being discovered and plugged long ago.
Is this inconvenient? Sure, but it's not a practical impediment for "batch jobs" like employment identity verification.
For "real time" things like renting a car, the vendors will have to begin the verification process, do things to "kill time" like review the auto-rental contract or go with the sales clerk to inspect the car, then tell the customer "you are good to go" or "we need more information to verify your identity."
For situations like a bank where you can't "fill the time," the bank can set up a highly-secure on-premises id-verification computer that can act as a long-term cache. Everyone whose ID isn't locally cached will need to "step aside" for a minute while their ID is verified. Or better yet, they can start the ID-verification process before getting to the front of the line.