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I guess people don't realize how short the time period is from when Dante is found by Kat to the final battle with Vergil. Dante doesn't have time to consider what or who he is yet, he is being forced along for the ride. When he says "the world is under my protection now, I don't think he fully understands what that really means. Obviously, he will learn the full weight of that decision if his story were to continue. And my mini thought process was supposed to be like what Dante was probably quickly thinking at that point since he was overwhelmed and everything had happened so fast. It doesn't mean that that's how he would think if took a breath and really thought things over (hence, sequel material).
I mean in the end, Dante literally says, "I don't know who I am anymore" and Kat tells him, "You're Dante, nothing more nothing less."
The missions after The Plan probably all happen on the same day, so think about the amount of insane **** that is occurring. Dante doesn't even get a chance to really think about the potential consequences of his actions. Hell, if you think about it. He's forced into it by Vergil: Dante's plan all along was to peacefully to exchange Lilith for Kat, but when Vergil shot Mundus's baby Mundus went berserk and started unleashing the destructive shockwaves. So, It wasn't Dante that caused the massive destruction it was Vergil.
After that mission, Dante probably thinks he has no other choice, but to attack Mundus's lair and take him out for good; pretty much Vergil the clever mastermind set up Dante to help him kill Mundus from the very beginning. Everything that happens during the game keeps Dante involved is due to the decisions that Vergil has made. For example, think about the Kat situation, I bet Vergil intended on Kat dying or being kidnapped (probably dying) because he knew that would fuel Dantes rage and conviction to get Mundus even more.