A copper coin should be on a white background and a silver coin looks better on a darker background. You’re correct in cropping the background out but what little bit of the background shows looks so much better if it makes the coin stand out.
I agree but I think you misunderstood my reply. I add a background to every coin image I make, I just don't do it while I'm shooting the coin. I cut the coin out of my initial image and discard the background completely. Then I add a new background of my choice to my isolated coin image in post-production. Put another way, the background in the initial image is never the background in the final image.