I subscribe to the perspective that brain injuries are the injuries that matter most in sports, so that's what I focus on in the "what's more dangerous" question.
I had believed that rugby carried a lower concussion risk because of the lack of padding/helmets changed player behavior, but it looks like that's not the case.
This[1] is one of the best analysis/discussions I found.
It references this[2] 2021 meta-analysis as the most expansive.
A few key quotes:
> This meta-analysis demonstrated rugby has the highest concussion risk, followed by American Football, ice hockey, and wrestling.
> Meanwhile, a 2015 study in the British Journal of Sports Medicine looked at 13 articles that addressed sports concussion rates. It found that rugby placed first, with a rate more than three times higher than ice hockey and eight times higher than American football.
> One paper, published in 2016 in the American Journal of Sports Medicine, looked at NCAA Division I injuries over three seasons for both sports. The authors found that the overall injury rate was about three times higher in rugby than in American football, and that a subset of this, the concussion rate, was about 2.5 times higher in rugby than in football. The authors deemed these differences to represent a "substantially higher" risk for rugby.
TL;DR: Rugby's probably the most dangerous for concussions and overall injury of any major team sport.
[1] https://www.wral.com/story/fact-check-biden-says-football-is...
[2] https://www.documentcloud.org/documents/23776916-concussion-...