I'm kind of shocked that there are "two sides" to this discussion. I think that the action taken by everybody in this situation was appropriate, the pilot's comments should be taken at least as seriously as a passenger who makes a 'joke' at the checkpoint where we all have to take off our shoes and be virtually stripped.
Flying is a different sort of job and not everybody can do it. There are very few jobs where you can affect so many lives so quickly, and pilots should be held to a much higher standard of physical and mental health than stockbrokers, computer programmers, or even ship's captains. There are millions of people who are affected by epilepsy (which in many cases is treatable) who are valuable, smart, productive citizens but probably not the best choice for a job where crisis involves multiple flashing lights. I understand that the vast majority of people with mental health issues are not a danger to themselves or others, and the vast majority of people who have some form of suicidal ideation are not going to be mass murderers. However, flying is a job where the consequences of putting the wrong person in the pointy end are severe enough to outweigh whatever moral injustice comes from discriminating against people with mental illness.
To answer the question about flying with somebody with treatable schizophrenia, the answer is "certainly not!" The drugs used to treat schizophrenia are precisely the sort of drugs that one should not take while flying!
Psychology is an inexact science, many would argue that it is not a science at all (disclaimer, my parents were psychologists and I studied it fairly seriously in college as a career path.) Perhaps someday we will know enough about humans to know when somebody who is serious enough about suicide to talk about it with friends is actually at risk of committing suicide (perhaps with a bunch of people unwillingly along for the ride) but the research indicates that we are not there yet.