Why do you think the scene where Sakura cuts her hair is important?
It’s important because it signified the beginning of her development from an ordinary girl, into a world renowned Kunoichi.
Kakashi’s first impression of Sakura wasn’t that great, and he was very justified in his opinion. She did indeed seem more interested in pursuing her romantic interests, rather than refining her skills as a Kunoichi:
And later on, Sasuke had this same thought.
After returning from the Land of Waves, he saw that she had potential due to the tree climbing exercise, but also noticed that while the potential was there, her skills were very lacking. So he was undoubtedly a little frustrated that instead of working to realise that potential, she was instead prioritising flirting and pursuing him. So, he very bluntly explained this to her:
And while Sasuke’s words did seem harsh, they were necessary in getting Sakura to understand the full scope of her situation. She had all the potential in the world to be a great Kunoichi, so why wasn’t she doing anything about it?
The fact that those words also came from Sasuke himself meant that they had the biggest impact on Sakura, since he was the one she was trying to impress. They really did cut deep, but I think that was the only way for Sakura to really take a look at herself, and realise that her input so far, at least in comparison to her teammates, had been insufficient:
Therefore, when she was pushed into a corner, and finally had enough of constantly be saved, constantly being protected, and constantly watching her friends from behind, fighting her battles for her, she decided then and there that it would be the last time she would ever feel this way again.
She cut her long hair that had almost been a symbol of her romantic pursuit of Sasuke, and was now determined to put her life on the line so that she could protect those dear to her. She had promised herself that for once, they would be watching her from behind, fighting to protect them. But most importantly, she had finally found her spirit as a Kunoichi, and the resolve to get stronger:
Which manifested almost immediately, and produced very impressive results. Not too long ago, she would spend extended time and effort making herself look presentable, and would faint due to genjutsus which in all honesty, weren’t even that bad. Now, she was willingly allowing herself to get stabbed with multiple kunai and shuriken at once, just so that she could catch her opponent off guard, and her efforts paid dividends.
This girl, who was “more interested in love than ninjutsu”, was now a Kunoichi fighting to protect the lives of those she loved at the cost of her own:
That’s why this scene was so important.
To use Ino’s very fitting words - it marked the point at which Sakura began blossoming into “an even prettier flower than a Cosmos”.