Mbappe was given too many privileges in PSG, and that is what has affected his attitude. At any time he wishes to leave, they are always quick to seek for a way to make him stay. He knows how important he is to PSG as they have shown, that is why sometimes his attitude even towards the manager of the club may be disrespectful. PSG should him go, the manager they have is not strong enough to control him. If it was Erik Ten Hag or Pep Guardiola or even Mourinho, coaches who have a successful career and do not want any payer to ruin it, they will be able to control him.
What PSG did because they were very afraid of losing their stars. if mbappe leaves, their fans will decrease, and sales will decrease. No Lionel Messi, no Mbappe. Who else could become an icon for PSG? Yes, that's about why PSG privileges and maintains Mbappe. But for me personally, it's better for him to just move so that his career will be better and he can win a title he has never gotten.
But now, PSG can no longer persuade him to stay at PSG. because Mbappe himself has a very big dream, namely being able to win the Champions League, he realizes that staying with PSG is a dream he will find difficult to achieve. It is proven that last season PSG had a luxurious squad but could not do much together in the Champions League. I think it's normal for him to be arrogant because he's still very young, and it's true as you said he needs a tough coach to change his character. The perfect trainer I think is Eric Tan Hag.