They used Wolverine, Kitty, and Mystique the way they did because they were trying to balance the elements from the original story with the universe they had created for the movies.
The original story was written in the 70s, so that was the "present" part and the future was i think in the 90s. Kitty was sent back in that story by Rachel Grey to stop Mystique from assassinating Senator Kelly. DoFP was actually Mystique's first appearance in the comics.
They wanted this movie to bridge the original trilogy with the prequels, so they couldn't very well send Kitty back in time. They could have sent Charles, Erik, or Wolverine, but the heart of the movie was Charles and Erik's struggle/friendship, so they sent Wolverine back. And they did a good job with him. He wasn't the focus at all.
I guess they made it so that Kitty was the one who sent him back because they wanted her to remain important to the story. It doesn't make much sense, but it was a nod to the comics.
And Mystique's important role wasn't a stretch. She was the one they were trying to stop in the comics too. But in the movie universe, they had established her as a stone cold killer and mutant activist, but then showed that she was like a sister to Charles growing up. This movie showed how that transformation happened in her.