I tried the simplify and got those thin outlines, too. As others had already said, there is really no need for the simplify operation if you're going to output it as a bitmap anyway. Just overlap your shapes.
It's an anti-aliasing thing. If you export as bitmap without the anti-aliasing option, you get a clean shape without the white outlines. Maybe you could export twice the size without anti-aliasing, then reduce the size by 50% to get a smoother outline? It's a 3D rendering trick some people recommend, too. (Rendering 3D images with anti-aliasing slows down the process, specially for lower powered pc's)...