Despite the dystopian setting it deals with subjects like loyalty, personal honor, self reliance, responsibility to others and self sacrifice.
It's too bad the first movie did not follow the book more closely. Although it was told from the girl (Katniss) perspective, she only survives because of the self sacrificing actions of the shrewd, humane and courageous baker's son (Peeta) who is her district counterpart
In the movie he's a wimp saved by the warrior-like Katniss, the book wasn't that way at all.
The subsequent books required too much suspension of disbelief for me, but my kids liked them and the new movie. I'll wait till its on TV
And Jennifer Lawrence although beautiful and talented is both too tall and too old to be a believable Katniss
This is, in fact, the "great man/great woman" theory at work---that there are millions of workerbees, but only a few Carnegies, Pattons, and so on.
I don’t know if the author intended it, but in the trilogy, Katniss is not so much the warrior hero, as the pawn who survives because others see her as a useful tool, and hope to use her for their own power.
She is a hero in the small-hero sense; she is self-reliant, she cares deeply for her family, and she is not afraid. But as you said, it is Peta who saved her family. Then he saves her, along with Haymitch and the game-runner.
Then Snow tries to use her. Meanwhile, there is a real rebellion, which she knows nothing about, but when the masses see her small acts of defiance, they gain the strength to stand. So the rebellion decides to use her as well.
Throughout the book, she knows little of the plans. She spends a lot of time sidelined. She throws in little bits of important help, but needs a lot of work from others to be useful to the cause.
And into the 3rd book, we see just how much she is being used. And while the leaders fear her, they had little reason to fear her personally, she was not going to be the leader of anything, just the figurehead.