"No flight school, in general, is responsible for anything other than making sure the applicant meets the minimum requirements (money and a Class 3 medical). It's not their job to do the [FAA]'s (or LBA's or FBI's) job. Sorry, chuks, those are our rules."
The last time I renewed my FAA CFI, using the AOPA/Jeppesen online course, there was one entire module about detecting people who posed a risk, teaching about the actions to be taken by the CFI and by his school.
Some guy with a history of severe depression might be just as big a risk as some clown named Abdul bin Talebani who shows up wearing a keffiyeh and who wants to pay for his flying lessons with crisp $100 bills. To take such people on regardless of whatever risk they might pose might mean that the CFI and his school are breaking the rules to some extent.
That module made it quite clear that "our rules" dictate that more be done than just looking for "money and a Class 3 medical." I assume that is part of what the complaint alleges.
By the way, it's DLH, not DHL: people, not packages!