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Joe, I know you have this feeling that drive wheels on an extended 3rd wheel pivot go flying around but I gotta say in 20+ years I've never opened a watch with a loose drive wheel and never had a comeback from taking one off and refitting it. The forces here are tiny, and just a little friction is needed. Hands are usually fitted much tighter.
As to cleaning up that pinion, it could easily go in a jacot with an appropriate carrier (drive dog) in a lantern, and polished up with either a thin burnisher or polishing paste on pegwood. It definitely needs some cleaning up. If you have a lathe, try chucking the pinion in the headstock, the a piece of pegwood in the tailstock, run the lathe slowly and hold the long spindly bit true with brass tweezers while you bring up the chucked pegwood. Should bore into it, then provide enough support to do the finishing work near the pinion. If the far pivot end is also manky (looks ok in photo) just twirling some pegwood down it should clean it up. Continue pushing the tailstock pegwood in the above example should work.