DIAMOND DUST with KY JELLY
I bought a small vial of the 1200 grit diamond dust from Kevin. You put a dab of KY Jelly (or a water-based lubricant) on the felt dremel bit (included) and dip some of the diamond powder onto it. I found that it's much easier to pour a little bit from the vial into the cap. The vial is too deep for the felt bit to reach the powder in the bottom.
Here's a decent but damaged coin from my HP experiment:

Note that the left side of the reverse begins dirtier. The roughness on the obverse is silvering.
Now, I took my trusty 6 micron 3 M radial wheel to the right side of both the obverse and reverse.

Reverse is noticably better... obverse only slightly.
Now, after the diamond dust on the left side:

The obverse is noticably better, while not much change on the reverse.
I tried a slightly different approach here- I smeared the jelly on the coin, dropped my rpms to about as slow as my tool would go, then hit the coin with the diamond dust on the pad, with adding another layer of jelly a couple times (every 10 seconds or so). I have a couple spots of copper shining through- but that might have been the silvering being removed, and/or no patina in those spots- but I feel it was probably the diamond dust. Coin seems to have been smoothed out a bit.

I then used the silver brush on the coin and hit it one more time with the 3M brush. The image with the black background shows the silvering much better, and is closer to what it really looks like.

The silvering is much smoother now, and in hand, the coin doesn't look all that bad.
This definitely warrants more experimentation-