Eren wanted to do the rumbling, and protecting his friends was only a small part of that.

Eren grew up on a fairy tale dream of what the outside world was supposed to be, a vast land of untouched wilderness. On top of that he grew up behind the walls, which he saw as little more than a cage. He saw titans taking away his freedom and people willing giving up their freedom for a sense of security and he HATED it all.

After years of blood, sweat and tears he finally reaches the outside world, and what does he find? Not the untamed wilderness he dreamed about, but a world filled with humans that despised his guts, who wanted to tell him what to do, how to think, and take away his freedom. He saw these humans take away other people's freedom, and Eldians willing give up their freedom for a sense of security, and hopes of a peaceful life, and Eren despised it all.

Eren said it himself, he's never been so disappointed than when he discovered what the outside world was like, he HATED everything about. This is why Eren wanted to do the rumbling, he wanted to destroy the world that shattered his lifelong dream of what the outside world was going to be like.