Labels, words, language in general came to be out of the need to pass on information from one person to another. If I did not need to express ideas, or if I had not received ideas (which would necessitate language to begin with), then language would be irrelevant. If it were for my own purposes, why would I require a common label with others?
Looking at the response on the no side, all I can say is that the expression of thoughts is, by definition, communication. We are used to expressing thoughts to ourselves, but if we had never done it in a language before, why would we start making labels? To tell them to ourselves? This goes against what we know about how languages develop. They develop based on USAGE. Word meanings shift and change due to predominant use, not individual choice. This is actually why people trying to engineer language (like with the debate over what 'gender' means) tend to fail. They are going against popular usage, and compromising communication, as they are using terms in a way that has no meaning for others.
I would take it a step farther and say that this development is what has led to our ability to self reflect, to communicate more clearly with oneself, and spur critical thinking loops in the abstract. One is a byproduct of the other, not a thing on its own.
And art? Really? Art is something made to express ideas. To other people. This is communication. It is meant to effect others, in stories, in prose, in exaggerated ideas for effect. All meant to spur a reaction in an audience. At its core, it is information from one person to another.
As for rhyming not contributing to communication, is it not clear that when things are catchy, when they rhyme well, people remember them better?
Language is to communicate. If I write this, I want someone to understand it. If I speak to myself, I express my own thoughts to myself, with the intention that I will understand it, at least.
Expression of one's thoughts is also communicating - against what y'man across says.
We wouldn't have language if it wasn't to communicate. To speak is to express an idea for the consumption of an intended audience (and sometimes an unintended audience).
If someone speaks a different language to you and they try to talk to you - or write something down for you to read - and you can't speak/read that language, it doesn't matter. What matters is that they *intended* for you to understand them. Whether you understand it or not is neither here nor there.
The primary function of language is expressing thoughts. When you think in your head, you often think in one language or another. I would know; I think in either Cantonese or English most of the time, and occasionally in Putonghua or French. When these thought processes happen, you are not using language to communicate.
Another purpose, although far less important, is as art. Language is often used as art. Chinese characters in particular have beautiful forms that will appeal even to foreigners who have no knowledge of the language because of their aesthetic beauty. Moreover, certain rhetorical devices like rhyming make poetry and other forms of literature sound better without really contributing to communication.