I don't think it likely that they would or even should give up on Youk in 12 games. I would want to see where he was in 10 more games. At 22 games you are well past 10% of the season. If he is still where he is, given the way Middlebrooks is hitting and the way Youk is not, and combined with the way Youk has slowed in the field, I would be thinking today about how I can get Youk some at bats, how I can keep him active enough to try to work him through this while having somebody else playing 3rd most of the time and then be ready to act on some plan by game 23 of the season. Maybe that is Middlebrooks, maybe you move Aviles to 3rd for some games.
However this team was constructed to win games with offense and it currently does not have nearly enough offense to do that.