Why does Weevil show a hidden heart of gold and act somewhat nicer to the player? Because the player isn't Yugi/Yami Yugi. Therefore, Weevil holds no grudge against the player for anything like Duelist Kingdom and does not even lose to him in a tournament, averting a great deal of humiliation. Plus, unlike the Pharaoh, the player does not (apparently) laugh at Weevil. Thus, we see his nicer side.
Reshef draws on the spirits of defeated duelists to regain its power and initiate the ritual for its resurrection. Given how many wins a player needs to get this far, as well as all the duels you likely lost, every duel you fought served to increase its power.
Fridge Logic
If the puppeteer from the manga and the mimic from the anime exist in the same world, how did the Reshef universe's Duelist Kingdom go?