jhauswirth kindly gave me some feedback about my file.

Grey zones were actually caused by a not-so-obvious z-fighting issue. Let me copy and paste jhauswirth's answer:
@unknownuser said:

Its the shadows that are causing the problem.
In SU in the "Shadows Inspector" there's an "On Ground" option. This causes the "On Ground" shadows and shadows from regular faces that are on the same plane as the ground to z-fight. So what you can do is turn off "On Ground" shadows, then right-click on your Scene and select Update so the new shadow setting is saved on the scene.

That works. However, LayOut images still lack antialiasing and I get bad edges. I still have to try out multiple line weight/resolution settings before I'm definitely satisfied with LayOut exported files...