I'm not defending the way publishers work - but I'm saying they would be highly unlikely to ask for proof of terminal illness. Other people have spoken on this thread about that being taboo. To ask your author would imply you didn't believe them, and you simply can't do that. If anything, the agent should be satisfying themselves about the general nature of the story before submitting it - assuming there was an agent here.
How are publishers meant to 'spend an hour checking out a diagnosis' without asking to see confidential medical records?
It's an extreme example, but I once worked on a book about a man who communicated extensively with a ghost from Tudor times through his laptop – non-fiction. There was no convincing way of getting 'proof'. You make a commercial decision to publish and if there are problems about veracity it is the author who is liable. If libel is involved, however, that's different. All publishers will get a libel read when necessary.
Where I do think publishers are at fault is, as the article by an ex-PRH employee said, in showing too much deference to bestselling authors. Once TSP had been a success, they certainly wouldn't have been asking SW any questions about the follow-ups, and they would blindly support their cash cow author, as they have done with their statement about her 'distress'.