> are incredibly few esports careers that could survive a Twitch ban
(emphasis mine)
People have never been entitled to a sports career, or to celebrity status by reaching viewers on someone else's platform. These platforms have democratized the "rags to riches" story, but whether it was the '30s or the '60s or the '90s, you've always had to play by the rules of the prevailing media empire. This is no different.
Companies are generally allowed to set arbitrary criteria for who they're willing to work with. There are some very specific exceptions around discrimination against a very short list of protected classes. Other than that, rules restricting conduct are usually legal. Note that I'm not claiming that rules like these are moral, I'm just pointing out that they're usually found to be legal when challenged.
The fact that you can sign up for an account in your bedroom in pajamas instead of signing documents at a lawyer's office in Hollywood doesn't magically make this a public square subject to a different set of rules.