Stainless steel is very difficult to remove all of the scratches but it can be done with a specific method.
I used to polish stainless for show quality concours restored vehicles and been in the business for 20 years, I just work in the office now.
You cannot skip any steps in the gradual increase of grits. the final belt we used was 150, then onto greaseless compound wheels, 200, then 240, then 300, then a soft, worn 300 wheel with a green grease compound applied to lessen the cut of the 300, then green grease on the buffing wheel, then white grease on the buffer.
The white grease doesn't do much in the way of cutting, it is just the 'finish' once the piece os buffed out with the green.
If you want to cut more on the buff than the green compound, get the black grease with the emery in it.
You must cross cut each increase in grit so that you can see the previous grit lines and watch them as they are cut out.
When buffing, it's important to remember that the grease is what does the cutting and it doesn't last long on the wheel, so apply regularly.