I think the objections are similar to those against Pascal's wager. Of the four quadrants, the "Believe in god, but god doesn't exist" lists an outcome of "no downside". But that isn't true -- from the perspective of a non-believer, there are downsides to living the life of a believer.
Similarly, your position seems to be "If I give a diagnosis of schizophrenic, but they aren't schizo, there's no downside" (the charitable interpretation of your motivation being "it's better to err on the side of caution"). What you are missing is that it is harmful to tell someone they are schizophrenic when they are not.
Edit: try to put yourself in their position and understand how they would feel. How would you feel if someone at a party pulled you aside and said "I think you might have mental retardation" or "I think you might have cancer". How would a mother feel if a babysitter told her "I think your son has Autism"? Even if they believe these things to be true, they aren't professionals and it isn't their place to give such a diagnosis, and it is inappropriate to do so because of the potential harm caused by an incorrect diagnosis.
Edit2: Also, your post didn't contain any actual help, e.g. "here's the number of a hotline you should call". It was just "Here's my diagnosis".
I think in all those circumstances, the delicacy of the delivery, disavowance of unearned authority, and presence of actionable advice makes the difference between "appropriate" and "inappropriate". I would not be in the slightest bit offended if someone at a party said to me "Listen, I'm no dermatologist, but that mole you said wasn't there last month looks a hell of a lot like one my co-worker had, and it turned out he had skin cancer. Couldn't hurt to get it checked out." Broadly, that tone was what I was trying to achieve.
While I don't disagree there's a negative effect from a "false positive", I think it's likely very small. The worst you'll likely do is offend someone. And while your Pascal's wager analogy is astute, I don't think the same objections quite apply - this case is a much more straightforward one of "high probability of very low harm, vs low probability of very high good", closer to buying a lottery ticket than Pascal's dubious infinities. Precisely calculating the expected return isn't possible, so you have to apply your best estimates.
My post did in fact contain actionable advice - "call your loved ones" - which has the useful benefit of working for many psychological issues besides schizophrenia, and also just being a nice thing to do generally. It was the best advice I could come up with. However, I can see now that in this instance I could probably have achieved the same result by giving the advice without the "diagnosis".
I think the objections are similar to those against Pascal's wager. Of the four quadrants, the "Believe in god, but god doesn't exist" lists an outcome of "no downside". But that isn't true -- from the perspective of a non-believer, there are downsides to living the life of a believer.
Similarly, your position seems to be "If I give a diagnosis of schizophrenic, but they aren't schizo, there's no downside" (the charitable interpretation of your motivation being "it's better to err on the side of caution"). What you are missing is that it is harmful to tell someone they are schizophrenic when they are not.
Edit: try to put yourself in their position and understand how they would feel. How would you feel if someone at a party pulled you aside and said "I think you might have mental retardation" or "I think you might have cancer". How would a mother feel if a babysitter told her "I think your son has Autism"? Even if they believe these things to be true, they aren't professionals and it isn't their place to give such a diagnosis, and it is inappropriate to do so because of the potential harm caused by an incorrect diagnosis.
Edit2: Also, your post didn't contain any actual help, e.g. "here's the number of a hotline you should call". It was just "Here's my diagnosis".