I have polished lots of glass , you want to use the firmest pad, my firm ones are "yellow" and I use the Griots fine glass polish, its not a fast process and use meduim speed 3 on a 5 speed dial and just keep repeating the process. it doesnt remove the scratch put polishes the inside of the scratch so you dont really see it as much. on out side windshiled glass that is really badly road oil and wiper blade contaminated i use a commerical degreaser, then glass cleaner then several applications of the polish then a final glass cleaning. you know you are done when you spray the glass cleaner on and you cant see the wiper tell tale radius.
I dont think the maker of the polish matters much but you want the heavy of the solutions if you have a choice.
interesting door soloution! i like it