After restoring the grills on my 65 Dearborn I can see why people would paint the large screen since the mesh is swaged into the pot metal vent.
If you don't remove the screen you will end up painting it black if you try paint just the sides of the vent. I think that's what you are seeing.
I didn't remove the whole screen, but just one side -- the side that is on the top of the included pic. That way I could slide masking tape in there to shield the mesh. The pot metal is very brittle so I pried the aluminum mesh out without prying on the pot metal. The mesh is kind of U shaped inside the swaged area so pushing it back into the recess is fairly easy.
I had no paint on any of my mesh although the disclaimer is still that I have a Dearborn car.