I suggest you test hand cleaner on a small area before using it on wood or plastic. I used waterless hand cleaner on a white board that had been damaged by permanent marker, and the marker came off, the white board was very clean, and undamaged. [I did make sure to try it out on a small area in a corner of the board first. That way I could be sure that there would be no damage.]
I like your hint because I have a very big bottle of hand sanitizer (got it on sale). Label says it can damage wood finishes and plastic. I guess you haven't had a problem with that?
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