No, it isn't. The fun is in the hunt itself, not the kill. Your assertion is like saying that orgasm is the joy and purpose of sex, but that's not true. And yet, without the orgasm, no act of sex can be said to have achieved its aim.
Nietzsche said it better: "Not every end is the goal. The end of a melody is not its goal; and yet: if a melody has not reached its end, it has not reached its goal. A parable."
and killing for fun is sick.
Why? It's not like we're talking about killing people. Killing people for fun is wrong, but only because it's people. It appears that either you are anthropomorphizing the game animals, or you aren't clear on why killing people is wrong.