The additional context is helpful, and I'm not trying to minimize your experiences. I'm just saying that an average guy who's played rugby and is a "typical man's man" can be inherently and unintentionally threatening to women, even if you personally have a friendly relationship with those women. Now some of this is likely cultural and country specific. I'm guessing from your reply times and mentioning rugby, uni, and camping that you're from Aus (or at least not the US). Most of my background is in the western US, so I understand that in your situation, things are probably totally different than my experiences. However, I have also spent lots of time in mixed friend groups, in traditional male-dominated areas (including rugby teams, interestingly), in very conservative spaces (as a very liberal person), and my experiences have been markedly different than yours.

I believe I’m fairly approachable as a guy and not threatening.

The issue isn't necessarily you. Women have been trained through long and often traumatic personal histories that men, especially traditionally masculine men, are dangerous. Add to that the fact that when you start to open up about emotional issues, your behavior starts to deviate from "normal" guy behavior. Not a problem, except that now you're an average guy (which usually means significantly larger and stronger than an average woman) who is behaving in unexpected ways, which means you're unpredictable from the point of view of a woman.

None of her perception or fear is your fault, but it is literally a dangerous situation from the woman's point of view.

Women are terrible at giving sympathy or comfort to men.

I don't necessarily disagree; I'm just trying to get you to think about whether that's because women don't care or because in nearly every culture, women need to be exceptionally cautious around men, especially men in emotional distress that might behave unpredictably. Even if they know you really well under normal circumstances, when you start to deviate from "normal" behavior, women need to be on their guard.

The “worst”, most “toxic masculinitiy” environments are better than the best women encounter

And that's where you lose me. Yes, "masculine environments" can be a great place to open up and get emotional support, but they can also reinforce harmful habits and act as an echo chamber (much like male-dominated internet discourse). I'm not saying that you should replace your male friends with female friends, or that you should stop talking to your male friends. I'm just saying that women can also provide that support in many circumstances (in my experience).

If you're consistently having bad experiences when you talk about emotional issues with women, then it may be the way you present the issues, the group dynamics, or the specific women that you choose to open up to. To say point blank that "The worst, most 'toxic masculinity' environments are better than the best women encounter" is where I disagree.