Honestly, him destroying the cards he doesn´t like seems pretty stupid. For all he knows, they could still prove useful in one way or another.
I mean, the first round already diverted from his memories with a freaking zombie-apocalypse happening thanks to the butterfly effect. And with different people surviving thanks to him acting different, the butterfly effect is gonna get stronger over time, making his knowlegde from the futur less useless with every round. He might think that the cards are useless because he didn´t need them during the last time, but he might need them this time.