If all life on earth was destroyed, there'd be one hell of a stable equilibrium, but probably not one many of us would like to occur.
If that were an even remotely-likely outcome, it would have happened. Life is extraordinarily good at surviving and evolving new equilibria.
Natural ecosystems can only be expected to be robust against perturbations they have faced regularly for a time, which usually doesn't include much of what humans do.
Meh, the history of life on this planet is one long series of massive, unexpected perturbations, ranging from ice ages so severe that the equatorial seas are covered with several meters of ice, to massive volcanic eruptions that block most global insolation for years, to massive meteor strikes. In addition, the ice core records show that the planet has undergone radical climate change (much faster and more extreme than what we're currently seeing) without any cause at all as far as we can detect, as recent as 60K years ago.
As long as we rely on nature to survive, we shouldn't scoff at the idea that our actions can have disastrous consequences on our own habitat.
Certainly. Equally, we shouldn't ignore the fact that doing nothing at all (assuming we could) will also have disastrous consequences on our own habitat. Earth changes all the time, in all sorts of ways. If we want stability we need to learn to actively engineer the planet.