I wouldn't be so quick to dismiss this idea. This would only effect less than 10% of regular season games. I wouldn't do this for postseason games.
There are a few things you can do here. You can remove any stats accumulated in extra inning games so as to not effect those numbers. Or you can come up with new stats, like extra inning inherited runners scored, or extra inning RBI, etc... Keeping them separate from regular innings stats.
If both teams play start with a runner on second there is still homefield advantage here. Lets say the away team score a run. The home team will know that they need to adjust game plan and worry about getting on base as opposed to moving runners over. Or they can play it safe and go for another inning. I don't see any disadvantage with it.
This is a culture shock to baseball so there is no doubt this would be unpopular but measuring this in the minors to see how grossly it effects games is worthwhile to see. It may have the effect of extending more games and in that case its a bad idea. It may show that better teams still win and it lowers innings played, which could be a positive thing.