Also, I don't mean that any of the games I mentioned were designed to teach me anything (I'm pretty sure they weren't). But they still did.
I see. I think this is a very important distinction to make!
Most of these so-called art-games seem far too much "on purpose" to me. Almost, indeed, didactic. The creators seem to have a kind of message or lesson, and then they will teach it to you. That's what it often feels like. I don't like being taught like that. (If I want to be taught, I want an academic context, I want it to be dry, hardcore, focussed. Not playful.)
But learning something form experience, that's a whole other matter!
And I agree that there's a lot of potential for this in interactive media. As designers, we should focus on creating opportunities for experiences. Not on broadcasting some message or teaching people the result of our own experiences.
I would personally use the word "grow" or at least "change" instead of "learn" in this context.
Give some examples! Discussing about things we do not make but we can make without any examples or ideas is not only dry but pointless. Talking about abstracts without trying to give it a imaginary shape or colour of any kind doesn't lead anywhere, really.
So, first of all I do not agree that art games are created "on purpose". Look at origins of The Passage (Rohrer) or Aether, Coil (2x McMillen) or Edmund (Greasley). I can't agree at this point. They all seem to be created quite spontaneously, be manifestations of feelings (death of old neighbour, childhood, mother's dealing with divorce) or something that the origins I don't know of (rape) but it definitely doesn't try to teach us anything.
I think that this whole "teaching" that you talk about is close to a story with a moral. Author can build a story because he has a moral or perhaps find it during writing. Or even do it unawarely. If the story is written well and told for some more reasons than just lecturing the reader about right or wrong, it is most likely that we will approach the moral the way we can think of it and the way it can inspire it. I would even go further and say that it is like idea that Aristotle had. Want to teach your kid something? Don't ever say "see son, this is how you NOT do it" but become an example for him. Do things as you want him to do and he will follow you. Perhaps it's the same with the story. If author is natural in his writing and the reason why the moral exists at the end of the story originates from his experiences and his behaviour, not theoretically created rules that written on paper looks terrific but he is unable to follow them in practice, then the moral appears much more appealing to us, readers.
Now, is moral exactly a teaching? I don't think so. Teaching happens in or outside of schools and is based on talking about certain subjects. There is always theory and it is sometimes followed by practice (I would say that history is highly theoretical but archeology will follow second path: theory followed by practice). That's what teaching is in it's most well-known form.
But morals appear in stories. Telling stories is not teaching. Stories that have hidden messages are not teaching. Is "Alice in Wonderland" somewhat political even though it was created for a young girl? Is "Winnie the Pooh" highly taoistic even though kids read it around the world having no idea about that? See, no one says "Do as Winnie!". Instead, book describes fascinating way of intuitive reasoning that Winnie practices. In an enjoyable way. And it is fascinating for both children and for adults (incl. those interested into Eastern Ancient philosophy). So it teaches! But only in a way. It teaches if you want to be taught. Message is hidden and it doesn't shout right into your face, right? It seems to be there not because the author had an idea of putting it there but because it seemed natural for him to write the character the way he felt he should be and act (Winnie).
Now, how can we know what a reader/player/interactor shall learn from experience, Michael? Honestly, I don't know. People are sooo different and they change all the time. For example, I've seen "Marley & Me" last year in the cinema. I found it touching like many other people but most valuable thing I have found in the movie was an image of a family of two adults and 3 children that were capable of staying calm and reasonable most of the times. I come from a country where people are stressed way too much and often happen to create much more problems than they really have. And this was different to me. Especially that movie was based on real characters. From the other hand, some people find the movie not even touching but boring. How can anyone know what a reader/viewer/player will feel and think?
I think that there is no way to predict it. We can do something the way we feel it and the way it makes us feel and only optionally keep our fingers crossed that on the other side of the screen someone shall feel/think the same. I say "optionally" because I really believe that a creator that is not an entertainer should care not at all about what audience thinks. That's the thing with being NOTentertainer. You have your reasons, you have your vision. Satisfy yourself, it's hard as it is to fulfill your vision and realize it. Why would you have to deal with people that don't even know you?! And shall never meet you. It also reminds me of Brad Bird saying (in making-of "Ratatouille") that he found out it is not really reasonable to bother with thinking what audience will think or feel if they will see it in 2 years. We make our (not)games faster but still, sometimes the very idea comes way before we begin to work on something. Maybe even 2 years before someone sees it. Why bother? There is like a chasm between how we perceive our work and how they can perceive it. All those saying how audience knows best is a bullshit for entertainers. Imho

I agree that we should create opportunities for experience. But I have found out that if there is a message (and message comes quite naturally if an author has something to say [generally, something to say]), the experience might be only much more valuable. Message doesn't necessarily has to be something strict and awkward, it doesn't have to be a restriction. It can be hidden behind the canvas, influence the canvas and be there but stay invisible for most of people. More precisely, for those who don't want to see or are unable to see (e.g. their world-view differs so strongly that they perceive things differently).
I find that message is a valuable thing, and that there is plenty of space for several messages in one creation. Simply, if one puts his heart, mind and soul into his creation, it appears and it is there. But, it makes creations much more personal and from what I observe most of authors do not want it or don't do it for other reasons. I find my creations empty if there is not me inside them, but perhaps it is just me.