sirpudding, I don't think you know much about how mental health records work?
I don't think that you know how background checks for security clearances work. They don't need her sealed records (although they can require you to provide them yourself), because they will have interviewed her parents, teachers and so on. Furthermore, this is for legal background checks and the Foundation isn't bound by law.
They certainly would have asked her if she had a history of mental illness, and she would have agreed under oath that the information that she gave was true and complete.
Besides this isn't addressing the problem of psychological screening. If she is taking medication for schizophrenia, the Foundation would certainly know, and if she is an untreated schizophrenic they would probably have diagnosed this. The Foundation has an interest in the mental health of employees, even more so than real life espionage agencies do. Not only can a Foundation employee leak, but they can also be possessed, mind controlled, infected with anomalous memes, ect.
If you took away a different conclusion than that, then maybe I need to reassess the email itself?
He comes across as extremely dismissive and the impression that I got was that he thinks she is somewhere totally mundane and delusional and because of this isn't going to do anything about it. He could track her phone. He could insert an MTF to recover her. He almost certainly should call the Foundation equivalent of a DUST-ONE (which should be mandatory). He could do a lot of things, but instead he sends an email to her supervisor indicating he doesn't give a shit about the people under his command.
(Also, I don't think the Foundation terminates people for incompetence, but that might just be a difference of opinion!)
The Foundation is an extralegal organization, and effectively a massive (and impossibly successful) criminal conspiracy. They can't just fire people for the same reason mobs can't just fire people. The options seem to be either kill them or make them into amnesiac vegetables (and the latter assumes a confidence in the effectiveness of amnestics that seems implausible).