Rock,
Two issues are being conflated here — is nature resilient and is extinction final?
Yes, nature is incredibly resilient. It bounces back after incredible assaults, like your Chernobyl example, or after, say, a major volcanic eruption. This type of natural regeneration can happen quickly — in years, decades or centuries — assuming the area is left to recover. It typically starts with early colonizers, that change the soil that allows other organisms to become established, which allow... It's not instant and it occurs in phases, but life does recover.
The second issue is extinction — the extermination of a lineage, race or species. We are undoubtedly reducing biodiversity globally. A century ago tigers numbered in the hundreds of thousands, but today the worldwide population is around 3,900. We've done the same to African elephants, lions, many primate species, frogs worldwide,...
Once a species' number drops to a certain level, recovery is difficult if not impossible. This is what I'm concerned about. Species disappear forever.
At a grand view, biodiversity does recover even after mass extinctions, but the time scales involved are measured in millions of years (not a human lifetime or two) and the recovery occurs without further setbacks. Image if after the comet or meteor that caused the Cretaceous mass extinction 65 million years ago the Earth had been continuously bombarded from space every ten years or so by another comet or meteor. That's the analogy — admittedly, an imperfect one — to what's occurring right now. Major assault > minor recovery > another major assault.
We're heading towards a biologically impoverished world. Maybe we'll avoid the worst of it, but I doubt that. Tiny bits of progress are offset by giant steps backwards.
Since we're all music heads here, the way to think about extinction of a species is to view it as though all traces of Beethoven's ninth were burned. At a grander scale, it's like wiping out all traces of Beethoven -- everything he ever composed. And given what we're actually doing, it's like expunging all traces of the romantic period.
Joe