Interresting topic.
It surprises me that at this point City can't make more out of counter situations, it's not like speed and passing skill is lacking at the team. If i was the manager of an opposing team playing against city i would however take the danger from a counter in mind, and perhaps let the backs keep back more so that a countering City wouldn't be able to exploit the flanks so easily. I'd count then on the idea that city is less likely to score when putting pressure as my own team has managed to get back, as replay's would kinda reveal simmilarly like these graphs.
On thing ive learned is that a team cannot be a one trick pony, the more weapons you have for various circumstances the better. If you specialise on the counter, opposing teams will try to nullify youre advantage there and force you on youre weaknesses, if youre strenght is in the air simmilarly opposing teams will put up more headstrong players in area's where they expect high balls to arrive. Optimally a team can be dangerous in any situation, they would have quick players for the counter, strong aerial players where it matters, a good long shot around the Attacking midfield, a great dribbler to make the action around the same area, some beast to dominate the opposing 16, a superb passer around the mdifield for when possesion must be retained, and besides all that the quality of having enough players that can move up when needed withought losing too much strenght behind which often involves having rather quick defenders that can pas around too. City however seems to have all that to some degree, and certainly not lacking in abbilety to control a situation you'd think. While i can see where the defense is lacking at times especially at the midfield, i'm left somewhat clueless as to why we have it so hard to convert when having control of the situation. Last game kinda showed it, though the 2 chances 2 goals at our defense was certainly the pinpoint of worry for me, we also had loads of control but we didn't convert that many chances from it, iam kinda at a loss where it's lacking too. If maybe one thing would seem a bit less of a quality here for us then it would be support from the back to the front and vice versa, which would indicate that we play a style that is somewhat further removed from the principle's of total football, with other words our attackers keep to the attack too much and our defenders to the defense too much, but that might be a very superficial analysis, and don't count me as much of an expert on these matters, just my 5 cents feel free to scrutinise.