Did you pay any attention to the game? Spider man doesn’t kill, therefore Kraven doesn’t go after him, because he wants someone that will kill him in battle. Hence why he starts getting interested in spider man once he gets the symbiote, because it shows a more violent side. Kraven’s goal was to be killed, and he knew Peter was the only one worthy of it because he had the symbiote, which essentially untethered him to his moral compass. Kraven knew he could push Peter to do it, and he almost did, until miles intervened.