When a team that is having good results changes coach, that team takes a lot of risks. The new coach may not be able to perform well, and the fans will put a lot of pressure on him. I think it will be very difficult for the new Inter coach to achieve the same feats or surpass the old Inter coach.
Whether they want it or not, it is not completely their control as it is a two-sided contract between a club and a head coach so that if one of two sides is no longer happy with the another side, they will separate with each other. Simone Inzaghi wanted to explore something new and on top of that we can think of salary attraction from Saudi Arabia but he has freedom to do this if he wants.
Inter Milan have to move onwards without him and it's only surprising for outsiders as they decided to move next with Chivu but not with any experienced coach. This decicion contains big risk for Inter Milan and they will have a very challenging season ahead by no longer having Inzaghi and working with a very young and inexperienced coach like Chivu.
Regarding Juventus, I don't expect great things, nor do I expect great things from Milan. Now, regarding Roma, if they had continued with the same coach, they would probably have been able to win a place in the European Champions League. With this current coach, I have some doubts.
All big clubs in Serie A likely have big changes from their head coach positions to their player resources and all those things contribute to a very chaotic, unpredictable but interesting coming season.