Here's what I do to polish:

If it was already polished, try hand polishing with a chrome polish and clean cotton rag. Polish with chrome polish, buff clean with clean cotton rag, repeat. If there are still scratches, remove them with wet sandpaper. Start with 1200. If that doesn't work, try 600. If that doesn't work, try 400. And so on. Then work back the other way with progressively finer paper until there are only 1200 grit scratches. Then get out the cotton buffing wheel and jewelers rouge or other polishing compound. Don't let the wheel burn the metal, when the rouge residue gets too thick, polish it off with a clean cotton rag and then start buffing again with the wheel.

You have tried just hand working a polishing with something like Simichrome right?

I think some of the opinions given are better suited for turnibg something originally “not shiny” into something “shiny”. Your thing is already originally shiny. A buffing wheel with rouge is the nuclear option for you. Don’t go there yet. (Except maybe that one spot you cooked)