Coaching and proper equipment are important. The biggest thing is that the better the equipment gets the more comfortable players are trying to hit harder and faster.
Rugby matches are very rough with near constant collisions but no pads. I'd be really interested to see a comparison study between to two.
I played 6 years of rugby. I also played high school football. My position, lineman, in football was a pretty non-collision position and I never experienced any concussions.
In rugby, every position involves collision. As a backline player most of my job was to run, but there was no way of ever avoiding multiple collisions in a game. Less impactful then football for the reasons you mention, but more frequent. I know for certain that I've had (at-least) a few concussions in rugby. For instance, getting up from a hit and having no idea where you're at for a second, and then feeling the world morph back into you as if it were knocked right out.
Concussions are a big problem in rugby too, and they have only relatively recently realised the extent of the issue. The NFL may have swept the issue under the carpet for years, but everyone was burying their head in the sand until very recently in rugby - there have been multiple cases of players clearly getting knocked out and playing on after a short head injury assessment this decade.
There are many accounts of players saying they don't remember huge periods of matches. Shonatayne Hape gave a great interview which is absolutely terrifying, where he describes the effects and how it became progressively easier for him to lose consciousness to the point where he didn't even have to been hit in the head by the time he retired:
Former UFC champion George Saint-Pierre has described in interviews how he basically loses consciousness for hours at a time, and doesn't know what happened. Like he'll come to and an entire day has passed. He suspects alien abductions (!), but there's no doubt this is brain trauma from all the hits to the head in MMA.
Concussion is a huge huge deal in rugby these days. Something like 50% of international matches see a player having to leave the field for a precautionary "head injury assessment". At junior levels players need to be immediately substituted if there is any suspicion of a head injury.
You don't hit opponents head with your head at rugby, so concussion is not so common.
The most frecuent problem is spinal cord injuries in scrum[1], which year after year leaves some players paraplegic.
Rugby matches are very rough with near constant collisions but no pads. I'd be really interested to see a comparison study between to two.