I don't think you're oversimplifying. They definitely failed on that front, and sometimes just plain did not show up for the game. I think my possession concerns might be more about his times before Dortmund. Gladback and Nice were both lower on possession than average under him, but I still think he would very much not be the guy we'd want.For example, last season possession stats were
1. Bayern 61%
2. Leverkusen 60%
3. Dortmund (Favre) 58%
4. Leipzig 54%
Generally speaking his teams have the ball when it's required (ie in the BL quite a lot) but he's not a possession for possession sake type of guy, who will also play a counter attacking game when the circumstances fits.
The problem I think was more in game management of those two things at times. His Dortmund struggled to shut down games (over simplification maybe?)