katniss was never going to convince snow that she loved peeta because doing so would be convincing him that lucy gray loved him. and he has spent over half a century fueled by his hatred of her, by her betrayal of him.
sotr makes it clear that snow is still affected by lucy gray, that he cannot forget her, and any reminder, any possibility of the return of anything like her must be prevented at all costs, which is why he kills lenore dove. to get to haymitch, yes, but she is also covey and a rebel, and killing her is an attempt to kill the lucy gray that exists only in his mind.
katniss was destined to die so snow could continue to live out his revenge fantasy against lucy gray. (the catching fire arena was meant for haymitch and katniss you cannot convince me otherwise)
and peeta, hijacking peeta, was another pawn in this revenge fantasy. do you think that controlling peeta allowed him to live in a fantasy where he and lucy gray were together? a fantasy where he had a glittering mockingbird caged to parade around that would sing at his command only. i do. and peeta broke that fantasy for him, he aided the thirteen, Live on Television, because snow could never control lucy gray. because snow has no control over anyone but himself. and people are not puppets no matter how much snow fancies himself a master manipulator and in control of his world, his country, he can never control everything.
snow was always going to do what snow does best. snow was going to fall.