I've thought this may be why American football has more of the full body collision, where two (very large and powerfully built) men collide at high velocity almost horizontal to the earth. They're trying to gain as little as a few inches in the collision.
I've heard complaints from Australian rules players who come to the US about tackling - that the way US players have learned to tackle doesn't have much strategic value and greatly increases the possibility of injury, buy they continue to tackle this way when they join Australian rules leagues.
That's what I've heard and thought about, I do want to be clear that I don't know too much about all this - what I've written is more of a question in case someone in the know could weigh in.
In American football you also may only tackle the ball carrier. You can initiate contact other guys (usually in the form of blocking), and that can be very violent at times...but I don't think there's a scenario where you can tackle someone without the ball (and it not be called for a penalty).
The main difference in tackling technique between American Football and both codes of Rugby is use of the arms. In Rugby the arms must be used to tackle the opposing player. A shoulder charge, with no arms used, is illegal and may result in expulsion from the game. In a recent New Zealand vs British and Irish Lions game, NZ player Sonny Bill Williams was given a red card and sent from the pitch for this exact offence.
The other main points to note are that the opposing player can only be tackled below shoulder level and, as you've pointed out, in open play only the ball carrying may be tackled (or blocked).
Tackling the ball itself and not the player.
I don't have stats to back this up, but my general impression over the last couple of seasons is that less scrums require resetting.
A few yards makes a huge difference in rugby, and players are certainly highly incentivised to give it their all, but even for the battering ram-type players, their ability to help the team actually get the ball across the gain line is a function of the stamina and mobility to get around the pitch for multiple phases of play, complex situational awareness and some degree of ball handling skills, not just the ability to block off the man directly in front of them for a few plays.
Not to mention rugby's increasingly strictly applied and penalised rules on what is and isn't a legal tackle which mean that a player who doesn't tackle in a relatively safe manner will be a liability to their teammates as well as themselves
There are just five men on the ice for a team in hockey, there's arguably more pressure for each individual player to "give it their all" at any one moment than in football where there's 11 men on the field.
As someone lucky to be from Pittsburgh where we have both the Penguins and Steelers consistently being the top teams in the NHL and NFL respectively, there is no shortage of serious brain trauma in the NHL. Just ask Sidney Crosby.
signed, this totally not still upset Seahawks fan.
(re: 2015 Superbowl)
Players are highly incentivized to "give it their all", and that's how people end up getting injured or receive massive head trauma.
Hockey can be just as violent, but it's a much more fluid sport, so the pressure isn't on as much for the extra exceptional effort.