Similarly we can't just assume troubled athletes who played youth contact sports, organized or otherwise, hadn't already suffered repeated head trauma which precipitated their various "demons," and increased their risk factor for engaging in compounding self-destructive behaviors, including alcoholism, drug-abuse and subjecting themselves to further damage by fighting repeatedly. Concussions aren't reserved for professional athletes, after all, or even adults. Their effects on the ability of kids to make healthy, responsible decisions, a struggle for even healthy children, I would imagine are that much more severe, too. Lack of evidence for causality in one direction isn't evidence for causality in the other.
It's a bit of a cheat to limit the scope to intentional suicide too, when "killing themselves" can be more honestly interpreted in the context of the discussion as engaging in any number of unhealthy or anti-social behaviors that lead people to early demise, intentional or accidental, or rapidly diminishing the quality of their lives, landing them in prison, etc.. To your question of suicide rates in particular, though, it could be that a majority of professional athletes who last in their sports long enough to accumulate severe brain damage also wind up with better support structures (coaches, agents/business managers, fellow players, possibly stronger or more closely-knit nuclear families, access to health care etc.) in place than the average person has at their disposal, that help offset the elevated risk of suicide, normalizing what might otherwise be higher rates. Not to say that's necessarily the case, only to point out that it need be examined and controlled for. Also, along those lines, some study I saw indicated a small but significant number of boxers (ten or thereabouts) die in the ring or shortly after a fight every year, which selects them out of the group who might later have attempted or succeeded suicide as a consequence of their brain damage, had they survived the immediate injury that caused their sudden death, which need also accounting for in the discussion. We can't at this time dismiss fighting-induced brain trauma as incitement to suicide without first factoring out elements that might be disguising its effect on a particular population anymore than we can link fighting to suicide without factoring out preexisting psychological or emotional problems, or other issues on that same population. So while we can't draw a direct link between a very particular A and a very particular B (maybe we never will), it's not like any of the data collected so far has done much to alleviate the growing suspicions either. Boxers in general exhibit significantly worse CTE than other professional athletes, which seems to at least convey that getting punched in the face isn't improving hockey pugilists' odds of avoiding brain trauma and whatever effects that has down the road for them. Certainly at this point it seems folly to insolently insist on proceeding from the negative assumption, that fighting in hockey is innocent until proven guilty. The suspicions alone should be more than enough for any reasonable enthusiast of the sport to reweigh priorities and demand the more brutal aspects of it meet some significantly higher threshold for inclusion than we've previously held them to. What reasonable justification can be put forth here to not start from the safer assumption, that letting these guys play russian roulette with each others' fists is stupid, unnecessary and unhealthy, for the players, and for the viability of the game itself?