Or as I would put it, if you don’t even know my wife’s name, or can’t remember that I don’t have kids, maybe start there before you ask about my mental health, cool?
Suddenly asking someone you barely know anything about outside of work about mental health, will be a shock, and quite frankly, will seem threatening more than it seems supportive. As in, “why is my boss asking about this? Does she think I’m not doing my job, am I about to get fired?”
I’m assuming that’s not how you want them to feel, so be thoughtful about how you bring these subjects up. By all means, work up to it, and work up toward helping your reports feel comfortable, but do it in a way that actually works, not in a way that makes them feel less safe.