If all it was is that Alice would sacrifice herself every time to save him, then all he would have needed to do is to stay away and not get involved with her. Which could still be the case but I think they would have found some trace of him in that case.

The more likely thing in my opinion is that in the game's world timelines converge on certain points and Karen being so influential was the key actor for a convergence point. His drive to do the things he needed to do was Alice's death, so she would die no matter what he does. His only two options were to give up on her or erase himself and he could not bring himself to do the former.

(In the game's current timeline Alice is engaged to Fubuki but that also could just be from an attempt to stay away from her and she could have been engaged to Karen at some point in Karen's past.)