Yes, but wouldn't it be obvious (or obvious eventually) that these healthy people clearly don't exhibit other symptoms of schizophrenia? Or am I forgetting something?
What about terrifying people with the thought they have schizophrenia but just haven't demonstrated visible symptoms yet? That concern comes up a lot with genetic testing because many things aren't certain but people panic and make significant life-altering decisions after a positive test result.
EDIT: … I should also have noted that humans are notoriously suggestible – with a mental illness it's really easy to imagine people both assuming every quirk is proof of a terrible disease or even starting to demonstrate symptoms psychosomatically after being told that this will happen eventually.
… or with family histories, etc. The real challenge, though, is probably just for doctors to be very open about false positive and negative rates and explain the issue directly. I think a lot of these issues would be easier if someone was frankly told “3 times more people receive positives on this test than actually have the disease“ instead of “it came back positive for [scary]”.
Nevermind the classic problem that was addressed in One Flew Over the Cuckoo's Nest and others: how do you prove you're sane? If the machine says you're insane, what way is there to prove otherwise?
You are assuming errors would be random, which is probably false. A false positive would probably be more likely for someone with similar symptoms, than for someone without any symptoms.
