When I was young I wondered why WWE (nee WWF) style wrestling wasn't in the Olympics.
What about Arm Wrestling or other "World's Strongest Man" style contests? I know they can be a bit cheesy, but they don't require expensive facilities or specific geography so any country could potentially excel. I think there's room for more one on one strength and endurance competitions. (and less syncronized ones)
Just for fun, check out what goes on at the Arctic Winter Games ( including traditional Dene contests):
For example, check out these Dene games like the Finger Pull