I have polished a lot of metals, mostly stainless steel, to mirror finish. Shafts I spin as fast as possible, 2500 rpm. something as big as a platter you’ll have to spin slower in the name of safety. Of course i’m talking about using a lathe.
The key to successful polishing is to work slowly up the grit spectrum. You can probably start with 240 or maybe even 320 if the platter has a good finish and no deep scratches. Clean after each stage as you move to finer grit. 800 or 1000 will give a mirror finish but polishing cloth (3000 grit equivalent) will really make it sparkle.
It takes longer then you would expect so be patient. Don’t try to do it in one afternoon. Polish for a while and go listen to music. Come back to it the next day.
Jerry