I agree with you completely, it has always riled me too when we hear folk say this, actually seems quite an arrogant thing to say, betraying maybe some arrogance of thought, a mindset not yet capable of thinking how we need to think, to fix the problem. Worse, that mindset is probably quite infectuous, so thanks for calling it out.
The truth is we don’t know if the Earth will be fine or not, we have never been in the billions, and there has never been another species in history which has changed the planet the way we have.
It might be that all possibility of life on Earth is already mortally wounded, by our hand (Our ingenuity applied destructively), and the only way to save it, is again by our hand (Our ingenuity applied creatively) , reversing what we have done to date.
In my opinion at least some of the damage is reversible. Ok we can’t bring back species already gone, and we can never undo suffering caused, but we might be able to create some more species, to fill in otherwise fatal gaps which might have already appeared in the natural ecological support system, but they will never be the same as were before.
But I think it is amazing how creative we can be as a species, if / when we ever get over the profit monster and unite as a species, almost the opposite of what we are when driven by profit; monsters, each and every one of us, we are like kids who have developed enough physical power to brutalise or kill our parents (mother Earth) , we just stabbed our mom, unwittingly, what should we do next?
Stab her again (“Its ok she’ll be just fine”)? Stab ourselves (“She’s better off without us”)?

Are either of those two options really in any way acceptable?

Or should we realise we are all actually in a very dire situation, requiring some really responsible action next.