Years ago, the National Association of Collegiate Directors of Athletics (NACDA) maintained and (occasionally) published catastrophic injury statistics by sport. Of course, "catastrophic" referred to fatal or paralyzing injuries, not "just" concussions. We know a lot more now.
The keys to reducing injury are equipment (like you said, some football helmets are woefully inadequate); coaching of the proper techniques; and eliminating the culture of violence that comes along with some sports.
You mention rugby -- a tackling, contact sport played largely without pads. Years ago, I played rugby at 6-1, 150, and survived many collisions with men 50-100 pounds heavier precisely because there was no deliberate attempt to injure opponents. (And those who did so were booted out of the league.)
Vis-a-vis soccer, rugby proponents often said: Rugby is a ruffian's sport played by gentlemen, while soccer is a gentleman's sport played by ruffians. While I wouldn't go THAT far, a big key to player safety in any sport is a common-sense, "sporting" approach to competition and fair play.
The American football culture is sadly lacking in this most fundamental regard.