I wouldn't consider any of those to be a normal person's everyday lens. They, from every use I can think of, are tailored to specific activities, such as portraitry, sports photgraphry, and other "zoomed in" activities. I wouldn't take a 70-2002.8 to Mount Rushmore for landscape shots just as I wouldn't take the 16-352.8 to a football game.
This site explains a bit of the reasoning behind the popularity of the 24-1054.0 as a general walkaround lens, but just because it's popular doesn't mean it will suit your needs. Personally, I would need to partner it with the 70-2002.8. I also would be frustrated with it when taking photos indoors in low-level light, which I do frequently.
It also mentions the 24-702.8, which is a much more viable option for indoor photography but leaves a bit to be desired in terms of focal length when compared to the 24-105.
The link above analyzes the performance of lenses, too, since focal length is not the end-all, be-all arbitrator of quality or usability.
I will say that if you're taking a lot of sports photographs and only casually take photos outside that arena, the 70-200 L series appears to satisfy several of your requirements. I encourage you to research your options there a bit further and compare it with your intended usage.
To end, like others said, a lens can only do so much. Make sure you take the time to learn how each lens behaves with your camera. It can make a subtle but important difference in performance.