These are not rectangles though. The projection has severly stretched many of the countries. Combined with the tilted angles, that should mean the corners of these "rectangles" aren't really 90°.
They used an equirectangular projection, because rectangles. But that means that some of these rotated rectangles are more like parallelograms when you look at them in another way.
Mercator actually would have been a good choice because it preserves local angles. Not sure why you'd be so opposed to it in this case. For this purpose, relative sizes don't matter at all.
And I don't think a local projection would clarify anything. Countries can get pretty big and non-local. If you strive for complete geometric accuracy on the scale of, say, Canada, the idea of a "rectangle" breaks down because there are no parallel lines on Earth, and four right angles don't bring you back to the orientation where you started.