"tl;dr": We don't include bot games on the 2P leaderboard -- what are the reasons for that, and why should they be different than 3P games?Agreed. Bots are extra-exploitable opponents, so they shouldn't be ranked, so we still shouldn't rank them in multiplayer.
For the joy of finishing out the game, the best option if someone drops is to replace them with a bot. The bot may decide the game between the other two players due to its decisions; there's no avoiding that, it's in the nature of 3-player games. You could have the player drop out completely - or have a bot that just passed - but that's not finishing a 3-player game, that's playing a 3-player game that turned into a messed-up 2-player game. I have played that game! Having a different player (in this case a bot) take over is better.
Replacing with a bot is the solution I would most prefer, as well.
Regarding ratings, I can absolutely see an argument for allowing the complete game outcome after replacement with a bot count. Currently, the bot is very likely to buy Provinces/points when able pushing the game to a conclusion. It's not unreasonable to expect far behind players in multiplayer games to do this to speed things along, too. Would ignoring the final outcome of games with resignations that are finished with a bot give better rankings than not counting it? I don't think that's an obvious "yes", at all.