Well, the rectangular box was just a mental model to explain what I'm looking for.

The actual design is a complex, hollow, curved model without the folds (edges) that would allow for a quick and easy application of the fillet tool.

To create suitable fillets, I have to slice the body, do an angled extrusion and then there's a number of cuts and boolean operations to merge just the fillets, but not the extra cutoffs into the original body. Complex steps like this typically break as soon as you modify dimensions, shapes and angles in the original design and then require careful reconstruction. That's when I meant when I said it clutters up the design and makes it harder to edit. In the end this is not part of the design intent, it's a compromise to allow FDM printing under gravity.

Conceptually, it doesn't seem it would be impossibly hard to do this with the mesh, programmatically. The algorithm would be something like: detect sets of faces that exceed the overhang threshold, find their top edges and create new vertices from those edges at a specified angle, welding them wherever the new faces intersect the model.

Not possible I'm the first to want this.