Just for skiing, f2.8 lens is overkill. It's expensive, it's heavier (I'm not going for heavy, as it's still super light compared to other stuff I use ), and it's bigger then f4.0 lens. And for outdoor sports, you don't really need 2.8 aperture. On the other side you mentioned hockey... I don't shoot some kid stuff, but from indoor sports it's always just top tier stuff (like KHL for ice hockey, Euroleague for basketball etc.) which means halls are super light, yet you still need f2.8 lens and relatively high iso (1600+). As soon as you go to lower level sport, it just means much much darker halls, where f2.8 lenses will help a lot compared to f4 lens. But it's up to you to decide, how much you will be shooting indoor, and if that justifies twice the price of 70-200/2.8 over 70-200/4.
Then there's another thing which noone seems to think of, when it comes to this. Camera's auto focus works much better with more light, so even if your aperture is closed to f16, auto focus still works at f2.8 (lens is closed down only at time when shutter button is pressed). So you get faster auto focus with f2.8 lens then with f4, and with darker places, this is quite important thing.
And for the end, forget about "just one lens" thing ;) It's not going to work, and it's not going to happen But there's also no need for whole closet of lenses. 70-200 is good on long end, but you will surely need something on wide end, especially for landscape. For normal holidays I normally take just 2 lenses with me... 17-40/4 and 70-200/2.8, everything else stays back home. I don't miss those 30mm between 40 and 70mm, but we are different, and this what I like is not necessarily this what you will like. :) So my suggestion would be 17-40/4 and 70-200... if 70-200 is f2.8 or 4.0 is up to you (look above). :)