Detect and remove overlaps resulting from polygons with crossing edges
A polygon layer with overlapping polygons (drawn on top of each other not adjacent and only contiguous for two edges (polygon 1 and 4 on the outer rim)) should have no overlaps resulting from crossing polygon edges (the overlap between polygon 1 and 5 and the one between 2 and 3, see below, should be eliminated and being merged with the bigger of the involved overlapping polygon). Overlaps between polygons 1 and 2, 1 and 3, 1 and 4, and 1 and 6 are to be kept as no edge crossing is taking place.
For context: polygons are meant to be in a print product where edges are set to overprint, so polygons overlapping are not spared out and edges crossing another feature will be visible.
Running the Remove Overlap (multiple) (Analysis) tool with the Thiessen method (or any of the alternative methods) will result in 2 valid features (below), while 4 have no geometry anymore, as the tool combines features that are on top of each other. This tool only produces the target result for overlapping features with crossing edges that are not fully contained within another feature in the layer.
In my post @BrennanSmith1 proposed a well working ArcPy solution. It would be great to implement this one into a tool of the GUI, so a more wider audience could simply use it.