So, I've used
Object -> Path -> Simplify. I thought I could easily go ahead and color the faces. However, there's still one problem:
There's not just one closed path for every face. When I drag the paths apart a little, you can see where they actually go and that oftentimes multiple anchor-points lie exactly on top of each other. I've tried to join, but this only works for the start/end of a path.
Is there any easy way to get the paths to form individual faces (rectangular shapes), so that I could color them individually? Of course, for this simple shape, I could redraw the faces with the pencil tool. I'm just wondering if there's a more elegant solution to this...
Note that - to my knowledge - Blender Freestyles don't support coloring faces when we still want to export as SVG, that's why I try to do the coloring in Illustrator in the first place.