I suspect in the situation, that most scratch removers will not fully do the necessary removals. The scratch removes / polishes are designed for clear coat paints on cars and are not designed to cut into the paint to remove oxidation like the old paint on the cars before the 1990s.
I've done scratch repairs on vehicles and I usually wet sand with a 2,000 grit paper. You could do this in an area a bit out of the way - down towards the bottom of the player as a trial. Keep the area moist - but, be careful not to get moisture down in the buttons, etc. Keep on sanding slowly without applying too much pressure. Then, wipe up the moisture and the fine sand / plastic particles with a paper towel. Then, use some automotive scratch remover - Meguier's, 3M, etc. With a very soft towel, keep working with small amounts of the polish and work in tiny circular motions. When done, wipe the whole area to remove any material or extra debris.
If your "test" area looks good, then move onto some other areas that are more visible to the eye - such as the viewing screen. Keep the area you work on small and focus on your work. Carefully, you'll really improve the situation and the looks of the player. You might not get it back to 100% or like it was on the first day - but, you'll be a whole lot better off than you are now.