Yet billions of years in, until we arrived, there was no failure to be sustainable. We're the first species to potentially cause a global extinction event (arguably we already did)

> We're the first species to potentially cause a global extinction event

There's evidence that cyanobacteria did it earlier, on an even more massive scale:

"...biologically induced molecular oxygen (dioxygen, O2) started to accumulate in Earth's atmosphere... causing almost all life on Earth to go extinct." [0]

[0] https://en.wikipedia.org/wiki/Great_Oxidation_Event

Isn't sustainability only that the system keeps existing? This is the 6th event, and even if this one is caused by us, Earth has gone through massive changes before.

The system will keep existing even if we erase ourselves (along with a long list of other species), but other life will keep on existing even after we're gone until our solar system ceases.

It's up to us whether we want to keep existing or not.

Earth has undergone lots of (short) periods of unsustainable ecosystems. They always crash and annihilate themselves. What is left has no choice but to gradually come into homeostasis, because the instability of too many aggressive species just leads to chaotic behavior and more crashes.

That's not true at all -- there have been countless problems similar to this, it's simply that in time life has rebounded, and that species without an "evolutionarily stable strategy" have died out. Here's just one example:

https://en.wikipedia.org/wiki/Great_Oxidation_Event

[edit]

A bit more info on ESS:

https://en.wikipedia.org/wiki/Evolutionarily_stable_strategy