Yaya isn't a defensive midfielder so why do people think he is? He's a deep lying play-maker. He takes the ball off a defender or defensive midfielder, turns and gives it to the forwards. He's the pivot around which our attack revolves. What Liverpool dd was ensure that, when he got the ball, he couldn't turn. If it's one player pressuring you then you can usually turn but when there are two then it's almost impossible and the chances are you'll lose it. So he just played the ball back to the defenders, who were then pressed themselves.
When it was clear what was happening, we should have brought Sterling, Navas and de Bruyne deeper so that they could take the ball off Toure quickly, as they can't have two players on everyone simultaneously. So when he gets the ball with his back to goal, there are options either side who are facing the Liverpool goal and can take the ball forward or play Aguero in behind their back four. That failure can certainly be laid at Pellegrini's door. Having Delph or Fernandinho alongside him to do that would have been even better. The problem with that (assuming Fernando isn't the one sacrificed) is that Sterling or Navas has to be, which means less cover against their full-backs going forward.
But even having said all that, their first three goals came from our players losing the ball when they shouldn't have done. Sagna for the first (I don't believe it was a foul and even if it was, play on instead of standing there waving your arm around). Then for the second, both Demichelis and Mangala should have made the ball safe instead of surrendering possession to a Liverpool player who wanted it more plus I think Kolarov could have got a tackle in. For the third, we seemed to have three players around the ball but they allowed Milner to come away with it. Then, for that Can backheel, Sagna was ball-watching and again more interested in trying to claim offside than track his man. So you can certainly ask questions about MP's approach but the players have to shoulder much of the blame.