Welcome to the forums and thanks so much for joining us. Full disclosure: I'm not a parent, but I know that this forum is full of parents in the same situation as yourself - so hopefully they'll jump into this thread or you can find theirs.
I'm sorry to hear how much you are all struggling with your daughters mental health. I am glad though that she is seeing a psychologist; is that helping?
I think that the Doctors suggestion may be different if they knew that your daughter was drinking at the time, but at the same time it is not harmful to cover all the bases anyway in getting a 'diagnostic picture'. The more that Doctors and psychologists understand what's happening with your daughter, the better they can be at helping her to cope. While it makes sense the psychologist would know your daughter more than the Doctor, at the same time sometimes a different perspective can be helpful.
It's only normal that you're taking on all this stress and honestly it would be weird if you weren't. You care about your daughter and your concerned for her so naturally you'd be wanting to do everything in your power to help. With your daughters counselling, is there ever family sessions? I know you said counselling on your own wasn't helpful but with family sessions it's much more holistic and about how you can help and be there.
Ironically, one of the best ways to be helpful is to step back - like you said already. Maybe that could look like trusting your daughter in the work she's putting in with the psychologist, or making some time for yourself as part of self-care.
I hope that this helps a little bit. Feel free to browse the supporting family/friends threads to find other ones from parents too.