I agree that a community that conventionally used a log scale like you describe to communicate confidence levels ("probabilities") would have advantages. It is not clear to me how, or if, this community can make a transition to that convention.

Perhaps a place to start would be by modeling the behavior you suggest, and adopting the habit yourself? (e.g., you say you "think" that starting to think about probabilities as integers on a log scale would leave us better off... what is that expressed in your suggested notation?)