with today's technology, it should be possible for the camera to detect the focus distance and the aperture and then calculate the DOF with those settings (for a 'standard' CoC).
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There are too many things to consider here for such a system to be effective, simple and accurate. In particular, print size would be a major consideration. The aperture required for good DoF on a postcard size print could be wider than that required for the same 'appearance' of DoF on a poster size enlargement.
However, I suspect the main difficulty most of us face when trying to be precise about DoF matters is measuring the distances of the far and near objects that might be considered to be sufficiently sharp. A distance read-out in the viewfinder, to the point of focus, would be useful
Another issue is the enjoyment of the activity of taking a photo. Do you want to spend the few moments you might have to capture the scene, the window of opportunity if you like, being preoccupied with the precise distance of near and far objects, or would you rather contemplate other matters such as composition and lighting?