OP, I have been thinking about your worry about this. I too am the parent of a very private kid.
I wonder if there is a way that you could have a conversation with your D in which you let her know how much her desire for privacy around her mental health is causing you to worry. Framing this as your problem (rather than hers) might help. You are clearly willing to step in whenever she needs help. You trust her ability and maturity to handle things on her own. But your love for her and conditioning as a parent is always going to leave you with a concern for her well being, and it’s hard for you to know if her silence is unwillingness to share or a tailspin.
Maybe she can agree to share more or what to share. Maybe it comes with a condition (like no advice.) But maybe you two can find a way to satisfy your need to be at the ready should she need it (which is really important) with her need to be more independent and maybe, she can feel like she’s helping you as much as you feel like you’re helping her.
Just a thought. Her mental health isn’t an issue for you but your worry, which might be manageable, is.