Like I just mentioned in another comment, they're also a way to get you back to work ASAP. Just about everyone NOT working a comfy white collar office job needs to be working in order to make money. Time off is less income, people can't afford that so they do what they need to do to get back to work.
Sprained ankle? Injured back? Headache? Broken bone? All things that people work through everyday with some NSAIDs because calling out sick means losing income