I'd get some sleeve bolts and run them all the way through a set of the existing holes. I'd be very surprised if the problem recurs. Use the threaded side on the bracket.
Drill completely through the door at the size of the female bolt (around 3/8"). Insert the female bolt from the exterior of the door, and the male from the interior (through the bracket). Ideally the female bolt length is just less than the thickness of the door at that point to provide maximum stability, as shown here:
Alternatively, get yourself a length of 1/8" aluminum bar stock as wide as will fit there. Maybe you can find gold anodized to coordinate, or paint it white.
Fasten that to the door using six or eight screws, depending on how the layout works. They should be chunky, like the closer screws, to bite well. #12 or #14 should do. Pre-drill the aluminum to the outer diameter of the threads so that there's no grab on the way through, and pilot the door slightly smaller than the screw shank.
Once that's installed, remove it and drill and tap for #12-24 or 1/4"-20 screws. Remount the plate and mount the closer bracket with them.