Discord, a private company that does not enjoy any sort of common carrier status, may not have any obligation to make up its own rules, but it also doesn't have any obligation not to make up its own rules.

It is under no legal obligation. The authorvis arguing that it has a moral obligation not to censor. Everything you say about 'common carrier' and 'private company' is only relevant to legal, not moral arguments. The author of the paper is arguing that open source projects SHOULD use something other than Discord because Discord is censoring when they SHOULD not. The author brings up no legal arguments.

And I'm saying it has no moral obligation not to censor, either. As a private organization, by default, they get the right to set their own standards of conduct. In the USA, at least, the law adds some restrictions in how this can be done, but only in limited ways.

I honestly have a hard time seeing eye to eye with the idea that private organizations have to let users of their platforms do whatever they want. It strikes me as representing a rather solipsistic concept of freedom. One could argue that the specific restrictions Discord is making are shitty. It certainly sounds to me like they are, and one could then mount an argument that people should steer clear of them because they are acting shitty. But doing shitty things is not necessarily the same thing as violating a moral obligation.

My main gripe here is that communication has become extremely centralised under the control of a few private organizations. A society whose Overton window is defined not by the community itself, but from a few network effect entrenched overlords does not sound like one that's conducive to free expression, alongside other things I value.