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That is not even close to true.

They stop you from using their services, that is it.

You have no right to access popular private servers.

You have no right to a conversation because everyone else has the right to ignore you if they so choose.

Again, private companies can not be forced to give you server space. That pesky first amendment thing and all.

Show me where you have a right to access a popular server that is not owned by you.

Show me where you have the right to have a conversation. That means that others can be compelled to talk to you.

Oh, but it is true.

If their service is popular, then the public conversation is to that degree affected. Clearly.