I am sure you have seen it before on high school, college and even pro fields and courts. You look out over a practice and see a guy with a perfect physique. He is fast. He is strong. He is quick. He can jump over Yao Ming. Then someone throws him a ball and you realize he would be better off having skillets for hands than the useless appendages he was doomed with.
When most people think of athletic ability they usually think of speed and power. Hand-eye coordination usually takes a back seat until they actually see someone in a key contest, drop a routine pass in the end zone. It dawns on them, then, just how important the ability is.
Think about all the sports that require you to use this skill at a high level. Sports that you could not make a junior high B team with, even with a 4.4 forty and a 40 inch V. Could a guy be any good at hoops with terrible hands? What good is a great vertical, if you can't catch the ball to throw it down, or you kick it off your foot when you begin to take it to the rack? What about baseball? Not much good there, unless Charley Finley was still around and wanted to employ you as a pinch runner.
You might not be any good at that either since people with poor hand-eye coordination, usually have a general overall lack of coordination. You might be fast but you would probably trip on your way to second. You might be able to play football at one of the non "skilled" positions, but the fast guys usually play those positions.
So you would have to have hands to go along with your wheels. My earlier point comes into play as well—you need to be able to remain upright while running no matter what position you play. What about golf? Do I really need to explain this one? NASCAR? Absolutely, not! That is what that sport is all about.
There have been some very successful athletes who have performed at a high level with deficiencies in the "traditional" athletic skills but who had off the chart hand-eye. John Daley doesn't look like a great athlete and probably would lose a foot race to a half-dead turtle, but he has incredible hand-eye coordination.
Tom Brady looked like he was wearing concrete boots when he was running his forty at the combine, but we know how successful he has been. He can throw a football through a Lifesaver from fifty yards.
Larry Bird, although not a terrible athlete in the traditional sense, took it to the next level because of his phenomenal vision and hand-eye coordination. Magic Johnson falls into the same category. Jerry Rice was overlooked in the beginning because he was considered slow, but he could catch a bullet before it blew by him.
Then there are the athletes who put it all together. They have the speed, strength, hops and hand-eye coordination. Willie Mays, Michael Vick, Mickey Mantle, Bo Jackson, etc. When an athlete has all the tools they are really fun to watch. They are the truly great all-around athletes because they have everything, including the reflexes and hand-eye coordination.
You might be able to be a decent athlete with just a set of wheels or great leaping ability. You could probably be successful in track & field or as a specialist in a team sport, e.g. Forrest Gump (who had to have another player catch the ball for him).
I just don't think you can be a great all-around athlete without hand-eye coordination. I also don't think you can be considered a great all-around athlete with just superb hand-eye coordination. It works the same way. You can probably be pretty successful at a niche sport like Daley, or in a team sport like Billy Butler. Butler is a very good hitter in the Bigs, but he has to DH and often has a pinch runner come in for him in key situations.
To be "great" you need to have all the tools, including, in my opinion, the most overlooked athletic ability: hand-eye coordination.