Your premise is flawed. The "downs-for-distance" system is fundamentally different from rugby's contested breakdown. In the former your intent is to stop the runner with as little gain in yardage as possible. So players commit entirely to the hit, using their bodies like bowling balls in order to counter the runner's momentum. Inches can mean the difference between keeping or loosing the ball. What happens after the hit doesn't matter as the ball is dead once the runner is down.
However in rugby, in rugby your purpose is to turn over the ball in the ruck which follows the tackle. Body position during and after the tackle is critical, as is the ability of the defender to stay on his feet to better poach the ball. This makes the nature of the contact very different than in American football.
There is also the issue of blind-side hits on quarterbacks and receivers -- hits which you don't see in rugby because of the nature of that sport's off-side law.
Or we could just look to history. In the years before helmets became mandatory, American football was primarily a college sport, where young men were being killed and crippled with frightening frequency. In 1905 alone, 19 players were killed as a result of on-field injuries. Dozens suffered permanent, catastrophic and crippling injuries through brain or spinal cord trauma. The situation was so bad that the American government considered banning the sport. Since then helmets have helped, but the incidence of concussions and serious spinal cord injuries in American football still exceeds that of rugby by a considerable amount. You still get between five and ten high school or college football players killed in the United States each year.
And @ Tim. You're flaunting your ignorance. Try being taken down in an open-field tackle by Richie McCaw, Bokkies Botha or Sebastien Chabal. They are equally adept at dealing out the pain as Ray Lewis et al. They just do it in a different way that doesn't involve driving their (helmeted) head into someone's kidney..