There is a plastic safe paint stripper available. The problem with these older grills is the porosity of them. Chemical stripping could be in the pores if you don't wash well afterwards. This could hurt your paint adhesion later.
Think of it this way, have you ever seen someone use interior paint on plastic trim pieces, only to have it flake of later? This is because plastics have either
Mold release agents (If fairly new) or years or oils and conditioners penetrated into them. In order to properly paint an interior plastic piece, you need to thoroughly clean and prep it. This means removing those contaminants.
Think of paint stripper as a possible contaminant, If you do not clean thoroughly. Dish washing liquid wash and a wipe down with rubbing alcohol usually does the trick. If you are quick and brave, a rag slightly damp with laquer thinner is awesome, but be careful. Then I follow up with an adhesion promoter (bull-dog) and then primer/base and clear if desired.
The quickest and safest way to remove paint from plastic is a gentle soda blast. But not everyone has that equipment laying around their house.
