Personally I feel trying to appease a certain community of people who have, either for the sake of a joke or peer pressure (probably not the correct word but you'll get the idea), already come to the conclusion that skial is shit is a waste of time and actively worsens the server. I feel it would benefit you more to take more feedback in the manner being done now, by talking and listening to the community that actually plays here, rather than a few loud people who talk about it online, whose play on the server and TF2 as a whole is questionable at best. Reddit is just one place, and if that specific community, as the reddit TF2 community is not THE TF2 community, doesn't like skial for whatever reason, I don't think you should implement changes simply to benefit them, bar the opinions of everyone else. Even if it's kind of a popular joke to hate on skial, more often than not, especially in the posts you linked earlier, they are either downvoted (showing a lack of people agreeing with them), or directly confronted and refuted (as seen in one or two of the posts). While there are definitely some people who hold bad opinions about skial, feeding into those kinds of people will only lead you down a path where you will work tirelessly and never please them, all while alienating your current playerbase.
Second, as someone people would probably rightfully consider toxic, I can say safely saying trying to moderate something as broad as 'toxicity' is a slippery slope in itself. the world is full of many people with many different opinions, and what is run of the mill speech to one person can be the worst insult to another. sure there are the obvious cases like telling people to kill themselves and so on, but past that, this rule would invite anyone and everyone to create their own idea of toxicity as a rule, and if someone's borderline definition gets accepted once, it can quickly create a precedent that leads to abuse of the rule itself as was never meant when written.
Honestly, I don't think skial's rep is as bad as you think it is. As far as I've been able to read, its just a meme in the reddit circle and rarely goes farther than that, and even then taking steps toward appeasing them would not change this fact. Even if for whatever reason you dismiss everything I've previously mentioned, this is the big takeaway: you will never change people's minds by sucking up to them, keep true to yourself and what you do and the people who matter will eventually come around.
