This idea immediately got me thinking about The Hunger Games and I absolutely love it (both the idea and the book/movie series). This idea feels almost dystopian in nature, one of those things that seems like it should be in a movie set in 2080. But, why wouldn't it work? I can't think of a single good reason why it wouldn't...
The sense of national pride that must fill Olympic athlete hearts as they fly the flag of their country must be unparalleled, to have that feeling as a "normal" person would be unreal. I think events like this would be really motivating and encourage people to maintain their health and fitness as you said. Would the events be randomly assigned do you think? Would the event types mirror those of the actual Olympics, or would they be tweaked a little to suite the average person?
I believe that companies would be jumping at the chance to sponsor these kinds of events and the entertainment value alone could be priceless.
A few small points that would definitely need to be considered are insurance (if people don't actively volunteer/consent then there could be a problem here), declining of invitations to participate (some people could flat-out refuse to participate, possibly to avoid the fame), prizes (would there be money involved?) and finally when/how often the events are held (would it be tied in with the actual Olympics?).
Either way, where do I sign up?