This always leads to an interesting debate and although it depends on what the definition of greatest is in this case let’s limit it to the classic Olympian definition of athleticism: speed, endurance, agility, jumping, power, coordination…etc.
Basketball is the first thing that pops in my head. I’m not going to argue one way or another, but for now that gets my vote.
Decathlon? 
Given the OP’s restrictions, you’re undoubtably correct. 
My first thought.
If we’re talking team sports, it’s gotta be basketball.
Even if we considered all sports, I still think basketball has the world’s most elite athletes. Not taking away from decathalons, but let’s face it, LeBron James plays basketball because he wants to get paid. It would be very interesting to see him (and Derrick Rose and Russell Westbrook and other great NBA athletes) train for and compete in Olympic events.
I’d probably go for American football or maybe rugby for the greater emphasis on pure straight-ahead running speed than there is in basketball. Danny Ainge was too slow for baseball, but he did okay in basketball.
Not a lot of jumping, but the physical demands of pro hockey require speed, endurance, agility, power and coordination, as well as strength (to at least tolerate body checks, if not dish them out or get into fights) and rapid decision-making skills.
A long time ago I read an article that assessed a large number of professional athletes from a variety of codes and rated them in categories such as strength, speed, flexibility, stamina, aerobic capacity and several others - the two scoring top were dancers and male gymnasts.
FWIW
I thought about hockey, but a certain (small?) percentage of their movement is coasting on the ice. That doesn’t happen in basketball.
Similar to the rigors and skills of hockey would be lacrosse. And no gliding/coasting. I suppose all the running with a stick sport would be about the same. Shinty, hurling . . . are there any more?
Given the fitness levels of hockey players, the fact that they are gulping air at the end of many/most shifts and then go back out and do it all over again a minute later, I think the coasting isn’t something that really takes away from the effort they put in. The top players are basically sprinting for 25 minutes a game. It’s really hard to compare, though, since I don’t really watch basketball. How long is a typical shift for a player in basketball and how winded are they (I honestly have no clue…do they even swap out in shifts like in hockey? I’ve never noticed!)
I just threw out hockey because I think it should be part of the discussion, and it’s kind of the only sport I know! I’m bored though, so here are a few thoughts based on little-to-no scientific knowledge whatsoever!
Given the other criteria in the OP:
[ul]
[li]agility - is there more agility involved in the direction changes and one-on-one play in hockey than in basketball? Skates and ice are inherently hard to keep balance on, let alone do the forwards and backwards manoeuvres hockey players do (watch Jeff Skinner someday…the kid was a figure skater and is just insane to watch). I can run around on a basketball court without training, but can barely skate at all. Does that gap in newbies close completely at the elite levels, or is hockey still something that needs more agility simply due to the skates+ice rather than feet+floor? [/li][li]power - I’d think these are about equal. Hockey is often about short, explosive bursts of power to go after the puck, outskate an opponent, go on a breakaway, but basketball requires similar effort and more jumping (though the amount of jumping over other players that occurs in hockey is surprising!). Distances are perhaps comparable, when one factors in the glide and frictionlessness of ice. Passing the ball and throwing on net doesn’t seem like it would use significantly more or less power than passing a puck or a slapshot. [/li]
Power in physical contact though; hockey wins hands down 
[li]coordination - kind of goes with agility overall, but I think just applying this to the manipulation of the ball or puck, hockey requires more hand-eye coordination than basketball. Just the size of the puck vs basketball seems requires more effort to keep track of where the puck is, let alone hit it. I can only go by my own very limited experience, but I have no depth perception and have never really played, but I can manage to dribble a basketball and throw a free-throw. I can’t even send a weak wrist shot into a gaping net 20 feet away, let alone manage a slapshot, tape-to-tape pass, deke, pass between the legs, spinorama, shoot top-shelf or five-hole or any of the other fancy stuff hockey players can do. Heck, even many NHL hockey players can’t manage a lot of that, certainly not with any consistency, but every basketball player seems to be able to dribble, pass and score somewhat reliably (or am I totally showing my ignorance of the NBA level of the game?) [/li][/ul]
I’m not considering goalies in this.
As I said, I only really follow the one sport, though, and most of my post is based on my imagination rather than facts. I’d love to hear counter-arguments!
They’re certainly the ones with the most superhero like physique (swimmers would count too).
Aussie Rules Football may have an edge when it comes to long-distance runners. The fields are huge, and players have to cover vast distances over the course of a match. Some random person on answers.com claims it’s an average of fifteen miles, though without any statistics to back it up.
At any rate it’s more running than in other team sports, with kicking and tackling mixed in as well.
I find that very hard to believe. The players in the most recent world cup covered about 7.5 miles in 90 minutes on the high end. Aussie rules football players are supposed to cover twice as much in less time?
http://www.fifa.com/worldcup/archive/southafrica2010/statistics/players/distanceandspeed.html
With 80 minutes of game time, that’s averaging 5:20 per mile. There aren’t many runners who can do that in a race running non-stop.
I really doubt you have entire teams who can do that while also being tackled and what not along the way.
I would be tempted to nominate lacrosse players, too. If you’ve ever seen a game (go to the final Saturday night, Calgarians! We’re set to take the league championship!), they basically do everything set out in the OP.
Well, there are breaks between quarters, and according to the Wikipedia article, each quarter actually lasts 27 to 31 minutes when accounting for “time on” when the ball is out of play. So that’s…7:12 to 8:16 per mile, with at least three chances to rest. So the figure is somewhat less unbelievable.
I admit these are not very reliable cites. I’ve just heard it’s a whole lot of running.
http://www.theaustralian.com.au/news/afls-hi-tech-future-within-reach/story-e6frgal6-1111113299998
That gives around 8 miles a game. The 15 mile a game number is complete bs.