I would like to share what I would have done to have Sakura's story hold more resonance with the story (at least in part 1). This is not my attempt to hate on Sakura, but simply share how I believe how I would have handed her story.
1. Have her family in a form of low position in the village.
The Village Hidden in the Leaves is a Shinobi village with seemingly many civilian families. It would have been interesting to see how these many civilian families work within within and contribute to village as a whole. Essentially having Sakura come from the lower spectrum could have placed her more as the odd girl out. A normal girl within a class filled with many clans' offspring. This would also add more significance with her friendship and rivalry with Ino.
I would have her need shifted to uplifting her family's status within the village for her and her family have a better life, and her goal for a major part of part 1 is to rise through the ranks as fast as possible.
There is still room for her development and she could still keep major parts of her personality. I also believe this would add credibility to her initial hatred of Naruto as she superficially see's his laziness within the Academy as insulting to her and the other genius on their team, Sasuke.
2. Have her mother die during the invasion of the Konoha.
While I'm not one for tragedy for tragedy sake, Sakura sorely lacks what every other single member of Team 7 has, pain. The pain of losing Sasuke hits her hard but she is the only one who never experienced personal loss. The closest one she experienced was Granny Chiyo but her final moments was connected with Naruto and Gaara.
I don't just want her to lose her mother to be sad for a while. It leads into my next point.
3. Have Sakura journey with Naruto to find Tsunade.
Sakura had no development during this portion of the story and she had nothing to do during this time period. I'd say she should have been in that room wit Kakashi and the Jonin and after Sasuke left, Sakura would have followed. The goal of this point is to have Sakura with Naruto onto find a healer for Sasuke, Kakashi, and Rock Lee. This way Sakura could have formed a relationship with with Tsunade and Shizune and take note of their skills and they hers. This would lead credence to her asking the Hokage instead of her own Sensei for training.
I'd have Sakura, Tsunade and Shizune develop a stronger familial bond as the series goes on.
Well... that's it. Most of this pertains to Sakura during part 1 because that is where I see she needed the most change. During Part 2 Is much harder for me to shift because doing anything major would likely take away from the other Characters.
Thank you for reading.