Imagine if it were Facebook or Twitter instead of just a video game. You could apply the same logic to say that Facebook is just a private company, and what they allow on their platform doesn't qualify as a curb on free speech. But those two companies effectively form an oligopoly on on-line discussion, and what they allow on their platforms has a huge effect on the country as a whole. Because their policies end up dictating what people are and aren't allowed to say on the internet, their moderation strategies should absolutely be held to a higher standard.
None of this applies to Turn 10, because a racing game is not intended as a platform for discussion, but that's a nuance that needs to be considered. The blanket statement, "private companies should do whatever is good for their business" does not capture that.