Ecological stability refers to a concept that is never realized. You can pretend it exists by ignoring the variation that you feel is unimportant. Over time, that variation will make its way into the areas that you thought were stable.
The reason I wrote my remark is the relatively recent uptick in "in nature everything always changed regardless of whether we exist, therefore we're absolved from everything that's happening with the biosphere"-style arguments.
Obviously on the face of it the premise is true, but time scales, operationalisation and causes/responsibilities matter critically there.
Islands of stability exist (like our homeostatic bodies, species that have survived for millions of years, old growth forests, etc) in nature. But whether one cares about (not unnecessarily, prematurely ending) any of this is a very different matter.
