"But is it really curing the underlying problems that consumers are complaining about?"
IMHO, it is having an effect (speaking on a personal basis, with an inside view). What you see with the Care team using Twitter is the tip of the iceberg in an ongoing effort to transform our customer care approach. There are of course other tools used, ranging from Twitter to web forums (incl BBR) to online chat - not to mention regular phone.
All of these different care channels carries its own challenges and each of them improve at a different pace. But it looks to me like the organization is making a genuine, concerted, and intense effort to improve and change, and it has some visible effects like Twitter use. And one of those effects is that customers can be heard more quickly and the organization has become substantially more responsive. These are all really good things. There is of course lots of room to improve, but the ship is turning and they've signaled the engine room to set full speed.
That being said, why does someone call Care? When something doesn't work or a customer can't figure out how to do something, generally speaking, which means focusing on reliability, usability, and what features customers would like (and would like improved/changed).
So that's kind of why I tend to hang around places like BBR and our web forums, as well as a number of people on my team. This is obviously not accidental.
A large part of that effort is to identify and resolve usability or other problems sooner, to open up our 'development feedback loop' to influence and input from customers, and to get a much better idea of how customers use our services, what challenges they have, what they really want, and more. It may seem pretty basic to interact directly with your customers and directly learn what they want, but as companies become larger that sometimes becomes more difficult to do. Luckily, this little Intertube thing helps break down those walls.
As to the posters, while sometimes you think you may just be seeing a post from a customer care person, that may not always be the case (it's not for me). It could be one of the engineers that runs the network backbone, or who runs the DNS servers, or the email servers, or who developed some new service feature that people are talking about, or a million other things.
We may not always be posting, but we're often reading. And for every person posting there are probably several other folks just reading and taking it all in. But know that people are watching, listening, learning, adapting, and experimenting with new approaches.