Out of the bazillion different posts weighing in on the "Air" bit, your thoughts seem to be the most coherent, to me at least -- if only because of the point on branding you raised. Marketing would want the "Air" theme to be consistent.
But the larger point is this: I think any user in a multi-mac household, particularly ones with laptops, can relate to the need for such a product. Apple wants us to want more and more media, be it video or audio, and for it to be available for consumption through as many Apple products as possible -- iPod, iPod Touch, iPhone, Apple TV, iMac, Macbook etc. All this content has to be stored somewhere, and for now, that has been on our computers as they hold, for consumers, the largest capacity. Video, particular of the HD variety, complicates that equation in that the file sizes are many times greater than say, your typical AAC song of iTunes. Combine that with the fact that an increasing proportion of Mac sales are MacBook/MacBook Pros where hard drive space is at a relative premium compared to a desktop and you can see where an "Air" server fits in. Perhaps this represents the pumped up Apple TV Businessweek pondered earlier this week. In any event, give it a much larger hard drive with its own master iTunes account and allow it to act as a hub to other computers on the network (probably finite just to please the content providers). Users will be able to navigate the iTunes Store on their television screen, with the ability to purchase movies and songs, or rent movies, HD ones, without having to get up from the couch. MacBook users will no longer have to be mindful of how much media they throw onto their hard drive. Instead of syncing to the Apple TV from the computer, the roles are reversed, and the user will designate only the songs, shows and movies they want on their computer from the Air Server. It allows the stagnant Apple TV to be reborn as a media server, ready to eat up tons of HD content, and allows happy owners to cast themselves into the role of programming director of their own mini-broadcast network. Perhaps something is in the air indeed..