That gives us 19 interglacials (10,000 years or longer) and an unknown number of interstadials (shorter than 10,000 years).
Mankind was present for at least four of the interglacials and many of the interstadials. The Mastadons thrived. The tigers thrived!
This latest situation, the Younger Dryass, was unique ~ the other periods of glaciation did not not stop and start up right away. They just stopped.
We are still in an ice age. But there are a lot fewer big cats around to stop our advances. Humans didn't kill off the cats ~ hunger killed them!
But it was only the last ice age that man kind started to spread around the world.
I will give you another example, Australia
The Mega-Fauna there went extinct 50,000 years ago.
Why did they go extinct there 50,000 years ago instead of the 5,000- 10,000 here in North America?
Simple because that’s when man showed up there.
Same with The Pacific Islands 30,000 years ago, Madagascar 2000 years ago, New Zealand 1500 years ago.
When ever ancient man showed up, the Mega Fauna went extinct.
“Humans didn’t kill off the cats ~ hunger killed them!”
The two aren’t mutually exclusive, since we are predators competing for the same food sources, and humans have proved to be more effective in most circumstances. The cats have had to retreat to places with enough food to support us both, or where they have a clear advantage over us.
Also, your theory doesn’t explain why big cats died out in the Mid-east and Mediterranean long after the glacial periods were over, but at the same time that human societies were becoming more advanced technologically.