re: Do you know how far the Kuiper Belt is? Yes.
I respectfully disagree with the idea that NASA would try to kill Cassini. It's not in their interest. In addition, they didn't do aerobraking, just standard rocket braking, to get orbit insertion.
I don't know whether momentum transfer from Titan would be enough to get Cassini out of Saturn's gravity well, but I'm certain that if it was enough, it would still not be anywhere near enough to get Cassini out to Uranus' orbit, at least on a reasonable time scale.