I think if someone seriously injured themself doing parkour or base jumping (or bare knuckle boxing) there's an implicit assumption that people partook in that activity knowing the risks whereas with a professionally recognised high stakes sport (where the athletes may also have contractual obligations to perform) then there need to be higher standards and awareness.
For instance why is head trauma more of a problem in NFL than rugby when rugby players don't even wear helmets
I'm also wondering if that's partially connected to exactly the fact that they don't wear protective equipment. In many cases it's of course definitely helpful to have protection, but since it can also make the risks less obvious, using heavy padding might also encourage heavier contact. If, on the other hand, none of the participants are under any illusion of safety, that could discourage recklessness and perhaps even encourage a sense of mutual respect.
Helmets generally don't offer that much protection from concussions. They do reduce the risks of other serious head injuries such as fractures, as well as more superficial injuries, but if helmets make people think they can take heavy contact with abandon, that might increase the number and severity of concussions.
Full contact but restricted martial arts also exist and mitigate risks by limiting the range of allowed techniques. Some of them do it in different ways than others. Judo is full-contact but disallows strikes, which makes it massively different than something with strikes. Semi-contact karate avoids the full-contact part, which makes it different in a different way.
Full contact practically always comes with risks, and full contact with fewer restrictions comes at much greater ones. Spectator expectations or other parts of culture may encourage the athletes to take risks and go to greater lengths than they otherwise would. But it's difficult for me to agree that modern society should have no place for sports that mentally healthy people actually want to participate in just because it comes with a risk of physical injury or intentional roughness. If they wanted to participate in a different sport, they'd have options.
In the end it's of course a matter of where to draw the line. Rather few people would nowadays want to allow fights to the death even if the participants wanted that. (Although, in reality, I don't think such a sport would get that many willing and actually voluntary participants either.)