Speaking specifically, based on what you describe (neither of us is fully informed, of course), the solutions are easy and cheap: Verify the number of bytes restored, the numbers of files and directories restored, and verify checksums (or something similar) for individual files.
If they had even the most rudimentary tracking or inventory of those assets/artifacts, the same technical problems would cause a much simpler and faster business recovery; instead, circumstances forced them to inventory something that they (a) possibly didn't have and (b) didn't know if it needed to exist in the first place, and (c) in a hurry, without preparation or adequate tools or people for that.
IT couldn't and cannot fix that - implementing a process may need some support from IT for tooling or a bit of automation, but most of the fix would be by and for the non-IT owners/keepers of that data.
As the linked article states, they restored the backup seemingly sucessfully, and it took two days of normal work until someone noticed that the restored backup is actually not complete. How would you notice that in backup testing that (presumably) shouldn't take thousands of man-hours to do?