When the ice melted and the oceans began rising human beings were forced to figure some stuff out ~ maybe killing the big tigers was one of them. Still, as the ice melted the zone of grass that fed the giant elephants moved North as well and the predators followed the prey.
Serendipitously a comet smacked into the Earth and froze the elephants and that took the predators with the prey leaving behind only the smaller cats like lions and tigers.
Human beings quickly learned to deal with the smaller cats and today they are almost extinct.
I don't put any stock in that timeline (1200 BCE) and I figure any culture that could haul around 100 ton rocks could handle the occasional smilodon without too much trouble.