Humanity on Earth is doomed, the planet will be fine.
Secondly, we're making the planet a lot worse for more organisms than humans.
If humans go extinct, human timescales cease to be relevant. Since we're talking about the Earth, reverting to geologic timescales makes sense. Earth 10 or 100 million years after humans may not be that different with or without humans.
(I agree, however, that the platitude is unhelpful. We're human. We care about the Earth on a human timescale.)
The next few hundred million years is all the Earth will get before it loses it's amazing uniqueness and turns into another boring rock.
Do you have a source for this? I assumed that the changes to earth was still billions of years away, when the sun finally goes supernova.