Kompany is too hot headed at times due to the pressure of the captaincy he feels the need to do the work of a full defensive line this leads to him charging around like a headless chicken at times and diving into tackles that more often than not he can't win and causing the team undue pressure
This is exactly my opinion.
Kompany loves the big games, he loves to be a leader and sometimes he tries too hard.
The best defensive players in big games go largely unnoticed because they deal with every threat that comes their way calmly and deal with the opposition with ease.
The game against Stuttgart is a prime example of exactly what you've posted.
The first goal was route 1. We didn't win the header and then it was all too easy. That sometimes happens and you have to win the ball in the air. I wouldn't blame our defence too much, though both Kompany and Sagna could have been quicker and stronger in reacting to the flick on.
The second goal was poor positioning by Kolarov. He got caught out and it was very easy for Stuttgart to then get a shot on goal. Too easy.
Because of that Kompany then came all the way across to the left hand side. If you re-watch their third goal we had three players on two Stuttgart players, they played an easy pass, got in behind and then it was Sagna v their attacker (around the penalty spot). So our right back was on the penalty spot and all other defenders were over on the left... Kompany shouldn't have been anywhere near that side of the pitch unless he'd drifted over as cover, and the other centre back then goes and fills his position. He'd been drawn out and left us exposed.
I think he's trying a little too hard to assert himself and regain form. I don't know where Clichy is but we really need him back in the side as he's usually better defensively than Kolarov. Getting the balance between defence and attack is vital for Pellegrini this season. A settled defence is also key. Mangala and Kompany are yet to flourish together. It would help if they were played with the same full backs for a run of games in order to settle.
Until Clichy and Zabaleta are fit and ready to play (I don't know if we'll pick South Americans for the first game) then we should be playing with a more defence minded midfield. We've enough quality in attacking areas for us to still break the opposition down. I'd be concentrating on getting form and confidence back in the defence first. If we start off the season with a few clean sheets then with the players we have we can challenge for the league.