Hello and thank you for your thoughts !

Criminal psychologist! Interesting! A Mindhunter type of fella?

Haven't seen the show, but I get the concept. I wouldn't say that my profession is something like it, given the fact that there's a lot of mythology on TV screens for entertainment purposes.
Usually the manifested issue by the client is not in fact the true latent cause of their need to go in therapy. Of course when it's an obvious circumstance like you said -

a depression caused by the departure of a loved one.

that's another case, but usually 'the issue' is on an unconscious level which takes a lot of time to uncover and fix.
As for your second scenario I don't think that any professional therapist would simply ignore any kind of information from their client. After all, the session is about the client not about the therapist.

As for you looking for more objective opinion on what is considered to be 'illness' I suggest you read the DSM-V :)

Thanks for your contribution !