There are 3 reasons why we place a high importance on the 10 yard dash
Sport is played in acceleration
While sprint tests such as the 40 yard dash are useful indicators of an athletes ability to accelerate and then gain and maintain maximal speed, rarely do athletes sprint for an extended distance like 40 yards in a straight line in an athletic contest. However, a short sprint like the 10 yard shows how good an athlete's pure acceleration is. And that IS how sport is played, in repeat bursts of acceleration. From stealing bases, to getting out on the break in basketball, to the average play in football, the better an athletes acceleration, the better they'll be in true "game speed". If you can't accelerate well, you won't have much on field success.
Great indicator of relative strength
Recently there was a video circulating on the twittersphere of Sumo wrestlers running a 100m dash. While sumo wrestlers are of course immensely strong, they're also comically slow as the video showed. This is why it's lazy to simply say power is the greatest indicator of sprint speed. Better said it is relative strength: not how strong you are, but how POUND FOR POUND strong you are. It's mass specific force into the ground that generates sprint speed. This is why overweight or soft around the midsection athletes aren't fast. The reason the 10 yard dash is the greatest indicator of relative strength is that if an athlete is overweight, his time will be slow. ALSO though if an athlete isn't very strong, that will also result in a poor time. Many weak athletes need to add weight and power to be able to run a faster time. A good 10 yard time demonstrates the athlete is at an ideal bodyweight AND has a good amount of power for his size.
Short to long
As Arizona Cardinals strength coach Buddy Morris says, if you can run a good 10, you can run a good 20. If you can run a good 20, you can run a good 40. The fundamentals of proper acceleration are the same skills in sprinting that an athlete needs to be effective at any distance. Show me an athlete who has poor technique in the 10 yard, and it's highly likely they're sacrificing speed while playing their sport because of that poor technique. The 10 yard dash provides an excellent situation to coach and correct technique.
For these reasons, while we test both the 40 and the 60, the sprint distance we place the greatest importance on is the 10.
*Key disclaimer for the 10 yard: utilizing a stopwatch for timing purposes is not ideal whatsoever. The difference between hand timing and a laser is typically at least .2 (reaction time), and sometimes even more (coaches bias/incompetence). It's tough to improve without accurate feedback, and for athletes to make the improvements they desire, the timing system used to assess them must be accurate. Even hand start/laser finish such as at the NFL combine is non optimal. Laser start, laser finish, hand in laser is the only way to get consistently accurate times.