1638: Humans establish a permanent settlement on Mauritus. Extinctions of the larges native species follow, the Coot in 1667, the Dodo in 1662.
1250-1300: Humans arrive in New Zealand, Megafanuna extinctions follow.
350-550 : Humans arrive on Madagascar. Megafauna extinctions follow.
1700 BC: Humans arrive on Wrangel Island. Extinction of the mammoth follow.
4 000 - 5 000 BC: Humans arrive on the Caribbean Islands. Megafauna extinctions follow.
ca. 14 000 BC: Humans arrive in the Americas. Megafauna extinctions follow.
ca. 55 000 BC: Humans arrive in Australia. Megafauna extinctions follow.
Also see places like Crete, Sardinia, Cyprus etc.
The fact is, when humans come to a place with animals that do not fear humans, we hunt them to extinction. This is not news, it has happened regularly and well into historical times. Any explanation that claims to show that the Americas was the big exception would need to demonstrate why the first settlers behaved in a manner so extremely different from every other human population hitting a virgin territory.
These were all populations that have managed to survive through a vast number of climate swings, sometimes extremely large ones such as ice ages and interglacials before, with little evidence of stress. And the number of additional factors you need to invoke to explain these things without humans causing it multiplies hugely with every new set of extinctions you need to explain away.
Yes, sometimes it takes a while. Especially in large locations where humans have to learn and adapt a bit, and where refugias can exist for a while. But we learn faster than creatures evolve, at least the big ones with large generation times. Hence, the tendency in these extinctions for larger -mega- fauna to be hit hardest.
And when you have new waves of humans with improved hunting techniques arriving… well look at the Tasmanian Tiger, the Great Auk or the Passenger Pigeon.
Like I said, this is an entirely historically attested phenomenon.
The one continent that kept most of its megafauna is the one where it grew up with us. Africa.
Yes, the Wrangel Island Mammoths had genetic problems. They were an exceptionally small and inbred population because mammoths were dead everywhere humans lived!