I don’t believe they wiped out the large predators because of danger. I think they did it for prestige. The ancient greeks would go way out of their way to FIND a lion to kill. Before firearms, killing such an animal was something to be admired. It was a feat of skill and intelligence as much as physical strength. I don’t know much about lions in asia minor. But I do know they have wolves there to this day. Wolves are the reason for existence of the anatolian shepherd dog.
Then people showed up who could kill them at will.
We call them Neanderthals ~ smart, stocky, strong ~ knew how to use fire and make stone points.
These pre-agriculture people deprived the large predators of game and reduced their numbers handily.
Later on Cro-Magnon people, like us, came in with yet more advanced technology and began a serious program of eradication.
Now while there were undoubtedly folks who thought of the prestige of killing large predators, there were many more who thought of how many more deer there'd be if they could get rid of the large predators.
Agricultural people had even less use for the large predators.