Second, you're being overly demanding, I think, even about "solves a problem." We could do a whole song and dance about the human need for fun, but that would be beside the point. When you say "It's fun, but I can live without it," you changed the terms of the whole debate. We started with "solves a problem," but tons of things solve problems and yet you can live without them.
Finally, I'm pretty sure that there is a real category "invented needs" (or "invented desires", if you prefer), and that lots of webapps fall into that category. The existence of the product creates the need (desire) after the fact.