Are you suggesting a sanity cross check pre-flight?
Paramedics establish if a patient is mentally altered at a scene by confirming they are 4x4. Meaning you ask 4 basic questions and verify 4 correct answers e.g. name, todays date, who is the President, do you know where you are. Any hesitation or spurious answer helps assess the individuals state of mind.
If I may, let me share some of my personal experience from the entirely different point of view - namely, from the point of view of someone who's been dealing with major depression; what's more important though, four of my close friends have attempted suicide, one of them succeeding (erm). Socially vulnerable people tend to stick together, sort of.
First of all, I am not a medical professional, nowhere close, so this is entirely anecdotal evidence; yet, in all those four suicide attempts, nobody has noticed anything unusual in the person's behavior just before it happened, and I seriously doubt that any "sanity test" in the form of questionnaire, long or short, conducted by a professional, layman or paramedic, would discover the immediate trouble. In my opinion, when people are saying like "oh, his or her replies were shorter and more tense than usual, we should have known etc etc" - it's all hindsight.
Yet, at the same time, none of these unfortunate events were entirely unexpected, and I'm fairly sure that a proper psychological screening - let me emphasize "proper", that's not the 10-question self-assessment form - would reliably detect significant majority of those who're in danger of attempting suicide, sooner or later. That doesn't make them unfit to fly; nevertheless, it probably necessitates certain precautions and more frequent and thorough testing, that, it an ideal imaginary world, could be combined with a help and support program (well, a proper support program, not the one which ends with a SSRI prescription). It's never going to happen, of course - rather, the society in whole is usually more than happy to hand the gun and bullets to those who're contemplating the idea of shooting oneself, instead of offering any kind of help.
Yet again, a point I should make is that in all those cases of suicide that I happen to know, it happened when the person in question was alone. Just saying; and I don't know the exact proportion of cases when the person wasn't alone. But, judging from what I've seen, it feels to me that a 2-person rule - how inconvenient, "knee-jerk" and what else it might seem - may actually prove to be quite efficient, and it's quite possible that this measure could have prevented at least the latest tragic event that is being discussed on this thread.