it’s getting better, but there’s still no inciting event.

You added some elements of backstory (family cases of schizophrenia). This could be something we learn in the movie that will explain something in the movie like the man is maybe a schizo. I don’t think this qualifies as a characterization. Who is this man? How can you emathize with him? The world “man” is too vague. In a logline every world is precious, so instead of “man” you should at least use another word that says more. Often it’s his job. “A succesful loyer”, “a dirty cop”, but the rule is that it must be something that fit with the story, possibly add some irony (A loyer accused of murder, a schizo who plays the doctor). Think about your character and your story and write something precise and appropriate. Like Syd Field (screenwriting guro) says “if you don’t know who is this man, who does?”. you must know and tell us in a word + an adjective.

The word “mysterious” can’t be used in a logline, it’s vague and unspecific (and very much abused in bad loglines – I used it too but I won’t do it again).

When you will include the inciting event (for exemple why the man starts investigating right now?) and a compelling character I think you can have a viable logline.