Understanding the basics of how the spray works (rendering the bear's sense of smell, his main sense of navigation, obsolete), do cougars, etc. rely on their nose as much as bears do?
Um... the spray works the same as it does on every other mammal - it gets in the soft tissues, ideally the mucous membranes, where it causes an intense burning sensation and drives away the bear.
Yes, it would work on a cougar, but first you have to see the cougar and have enough time to deploy the spray. Which you probably won't since the cougar's main method of hunting is stealth. I'd bet you have, if you hiked any miles at all, walked right under a curious cougar or ten.... Far more of them have seen/smelled you than you will ever know are there.
"In the beginner's mind there are many possibilities. In the expert's mind there are few." Shunryu Suzuki