Recursive editing of boolean objects.

I like the idea, and I think it's possible to be implemented, but in my mind it would have disappearingly exotic use cases, while brings so many obscure implications.

1. When you move a face it could go through another face of the input mesh, so that the manipulated face can split up or disappear - how would you handle it in UX/UI?
2. An edge can be from an intersection of two planes from different objects, a vertex - from an intersection of three different objects. Moving this vertex will change three or even more objects simultaneously.
3. In your example it is easy to imagine how input objects should be changed - all the faces are perpendicular to each other. But what if it's not the case - input objects don't form straight 90 degrees angles. In this case if you move a vertex, it's motion isn't orthogonally decomposed into motion of input faces. The edit becomes not so user friendly - again you either have to snap the motion to either of input faces or invent something else that will be understandable by the user.
4. Should you invent a special, novel edit mode for manipulating the result instead of manipulating input objects? How would it interact other modifiers in the stack? How can it be made compatible with geometry nodes?

I think you have to think through all this cases first.

What I'd prefer is a "cache geometry modifier", that burn ANY previous modifier into real geometry that you can edit. If needed, you can disable it, change previous modifiers, and then recalculate the cache.

you can select both recipient and boolean object and enter edit mode. If you have the "geometry icon" enabled in the boolean modifier, changes should be realtime. The solution you propose, while clever, would raise many design problems: when you select that boolean-result face, blender should know that only a limited subset of operations are allowed....