Anti-Sakura have interpreted the "promise of a lifetime" as Sakura placing the burden of saving Sasuke on Naruto while she doesn't lift a finger herself. However this isn't the case!
During chapter 183, She didn't come to rely desperately on Naruto by making that promise. No, in fact, she came to join the team to save Sasuke.
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Right off the bat, Shikamaru stops Sakura, knowing her intention. He realized that the task of a bringing Sasuke would be difficult, as not even Sakura was able to make him stay. So, he cuts her off apologizing that she can't be brought along. She hadn't been allowed to join the mission as Shikamaru said: "Sorry, but I can't take you to the mission."
And she couldn't head out on her own. As that just wouldn't only be reckless, leaving the village borders without permission would make her an rouge ninja. Making matters much worse!
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Which is why it came down to this. She begged Naruto, because she knew she couldn't do anything.
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In the end. Sakura hadn't went to leave Naruto to carry this burden of rescuing Sasuke. She wasn't allowed to come along. Knowing she couldn't do anything she begged Naruto to save him as she couldn't.
