We've been hunting species to extinction for 60k years, yes, but we haven't reached global transformation of the environment until the past few hundred.

But you can also interpret "currently" on a geological time scale.

Hunting anything near extinction is very solidly at the last inch of the timeline encompassing the last 60k years.

Most extinctions, sure. Some think humans hunted Australia’s megafauna to extinction 50,000 years ago:

https://earthsky.org/earth/early-humans-wiped-out-big-animal...