Stories of countless species hunted to extinction show that this is not what happens.

Let's look at the physics(?) of the economy: where does wealth actually come from? It's either from transforming things into more useful things or increasing efficiency of processes. There are limits to both of those things.

We live on a planet that's in a rough equilibrium in terms of both matter and energy.

So we have a finite maximal amount of value that we can extract from a constant amount of matter - which means that overall wealth is limited - we're just moving it around and at a finite rate at that - somebody is getting a bigger slice of this pie than the others.