I don't remember ever thinking much about big cats. A zebra bit my foot as a child, but no cats
As for dealing with cougars on a practical level, scaring them out of one's back yard does little good when residential areas are expanding so much that the big cats and other predators eventually have no where to go but someone else's back yard. So what to do? Either shoot on site, or create a protected area for them. It's apt to be a combination of both.
In bear country people are expected to carry a powerful gun, but they're fairly safe as long as they don't get too close. Maybe some people will have the skill/evolution level to be able to safely live around cougars, but that doesn't work for most people at this point (or at least it's not perceived as worth the risk).
Likewise I have old hippy mountain friends who have no problem handling snakes, poisonous or otherwise. Still, if they come across a deadly one, and they have little kids around, most are apt to kill it, though they may apologize before hand and say they're only doing it to protect their children (my father did this).