The Hammer has a good point. Records show unprecedented rates of megafaunal extinction shortly after hominins showed up in any numbers. This happened all over the world.
We may not have actively hunted some of the larger or fiercer fauna, but we apparently made enough of an ecological impact to redound all the way up the food chain.
Then there was climate change. Once a secure ecological niche is established, animals often become exquisitely well adapted to it. Such overspecialized animals might be extremely well adapted to their lifestyles, but make some minor change, and they can't adapt quickly enough.
Smilodon died out at the end of the last glacial period.
Or was it a Pleistocene pandemic? Maybe an outbreak of Calabrian cat covid.