Maybe the term "communication problem" isn't the best. A better one might be usability, or interaction design. Good design has heaps of it. And it's not aesthetics alone. A designer should go a lot further - scope includes accesibility and (if you're unlucky enough to work with me) information architecture/design.
Knowing how a colour wheel works doesn't solve problems like a custom-designed, ticked-out über-awesome CSS -based drop-down list that doesn't respond to keypresses.
Knowing when to use search, and when to use tags, or how to group search results in a way that makes sense to the use context is worth paying a lot more for than just coming up with a cool UI concept.
The only point I am trying to make is that there are many many great webdesigners out there that can make beautiful websites that communicate well. But it it's often the same guys that design something like an interface or a mobile app.
The creative agency world is filled with people able to do artistically stunning websites but they are approaching it from a communication/advertising point of view where you might be exposed to the site once or twice. This is very different than doing something like say hipmunk.
Designing the actual interface/service/functionality/product is a different problem all together.