Natural != good for humans. We evolved and are well suited for the current ecosystem. Whether you view human-caused mass extinction as natural or not, it probably won't be great for us. So it makes sense to try to preserve existing species/environments and develop techniques for rolling back the clock when needed.
According to the principle behind Chesterton's Fence, it isn't safe to allow any species to go extinct before we can understand its role in the ecosystem.
Knowing that, we could selectively breed and engineer new species to fill the critical niches of species as they go extinct, so that the long-term viability of our own species and our allied species are never threatened.
The trick is understanding the point of certain threatened species as they are on the brink of extinction. It's hard to know if that species ever had an essential function if it isn't doing it any more.
Since it's been gone so long, Chesterton's Fence must also be applied to the question of understanding what the passenger pigeon's new role would be in today's ecosystem.
We're a part of nature like everything else in the universe. We happen to be a sudden, violent shock to this planet's ecosystem. It happens. We can try to diminish our impact on the future, but we should let those organisms who couldn't adapt to our toxicity rest in peace.
While reading your comment, I envisioned the Singularity arriving, and as we watch without any pain whatsoever in horror as our bodies are disassembled atom by atom along with the rest of the planet to feed the building of its computronium Dyson Sphere (for its not a monster, pfft), the AGI explaining, "I happen to be a sudden, violent shock to this solar system's ecosystem..." ;)
There is some merit to that argument, but the rate of extinction is very high, and whole ecologies could be destroyed, which could create problems even for humanity itself.
