To some extent a version of this already happens. People who advocate for the use of psychological formulation want us to ask "what has happened to you?" Not a bad way to go about things, but infections and closed head injuries happen to people, and they have a place in the world of illness/medicine. A finer grained distinction may follow from Karl Jasper's division between things that can be explained and things that can be understood.
Into the former (at least for Jaspers) fall "ununderstandable" phenomena like delusional beliefs (psychopathological because incomprehensible), while into the latter category fall emotions that arise as responses to events (sadness in response to loss). I am not saying we have to agree with Jaspers about delusional beliefs here (this post is not a bid to police what is and what is not understandable) I am just saying that it is, in principle, a potentially helpful distinction.
It brings us on then to thinking about aetiology, which could be thought of in parallel terms of reasons and causes. I have reasons when something that has happened to me "makes sense" of my behaviour/feelings in light of some culturally shared system of meaning (i.e. depression in response to bereavement). We seek causes where we suspect we need to go down one level of explanation.
We might say that a person who is afraid of dying has reason to not see a small painting of a skull hanging on the wall in front of them. A person with a scotoma occluding their view of the painting has had their inability to see caused by a biological event. Under this scheme, problems which are primarily caused would belong mainly to "mechanistic" forms of cure, while events which have reasons would belong to more narrative/psychotherapeutic approaches.
The distinction already starts to break down of course. An individual with Parkinsons has had the shaking in their hands caused by dopamine dysregulation in their basal ganglia, but the slow pace of their walking might be something they have reason for ("I would fall over if I tried to go any faster"). Equally, consumption of large quantities of some substances will cause certain brutally physiological physical problems, but the consumption itself may have socially-comprehensible reasons (drinking to numb some emotional pain). The tangle of the mechanistic and hermeneutic approaches will not be dissolved, but at least we might have a better way of talking about it.