Now, I certainly believe that climate changes at the time also played a role: for all we know, these might have even been responsible for most of the animal population reduction, through shrinking the habitat etc. But without the humans killing off the rest, they’d have survived, just like they did during climate changes for previous millions of years — it’s not like the climate just suddenly became unstable recently, after millions of years of high stability.
But yes, the overall point is that humans have produced cultural technology that made us just too good at hunting for it to be sustainable source of calories for majority of population, and this has already been true for many millennia.