The episode is not about saving Earth. It's not about rediscovering humanity. It's not even about mourning our loss. The loss of Earth is a background detail, and its point is that you don't matter, your time is at an end, and you just have to deal with it if you want to be part of the future.

You're looking at the episode through a "modern day lens", as the kids would say.

The episode is used to establish that the Time War has greatly impacted the Doctor. He is much more cynical and bitter, while Rose is depicted as utterly naive. The Doctor wants to rattle Rose's cage, so he takes her several billion years into the future to show how Earth will basically be nothing more than a glorified building demolition. He's lost Gallifrey, his home, and eventually Earth will also be gone.

But the purpose of this is also to show how Rose will gradually soften the Doctor as the series goes on. She is naive, and she does have some harsh lessons, but she also makes the Doctor realize that as tragic as his past it and that nothing will last forever, it can make what you do have now all the more precious.