We mostly agree, but would probably come to a semantic distinction: I think the Playstation and Xbox as platforms are gaming first but which also do other media on the side. iPhones and Android phones, by contrast, are phones first which also happen to be able to play games.
It's a matter of emphasis, form factor, and cost. I think, broadly, that the Google/Apple approach has been a winning one for convergence devices for some time, while systems which are gaming-first have not been especially successful at convergence generally. I think this Amazon box is the first console to even vaguely fit that Google/Apple paradigm; it is a multimedia box that also happens to play games, not a game device that also happens to do multimedia. I think there is a considerable portion of the home console market that would prefer the latter to the former, but we'll see.
The Amazon Box is the first real attempt at pushing the Google/Apple paradigm to the set-top box realm in regards to games. However, Apple's, Google's, and Samsung's smartphones and tablets tend to be best convergence devices in the portable realm. If that is applied to the television, the video game console as defined by Sony and Microsoft is likely the best convergence device since it plays disc content, streams and plays digital media, as well as play the majority of games on physical or digital media.
Virtually every analysis I've seen has predicted a contraction of the console market this generation. Maybe they're wrong, but I don't think it's likely the PS4/Xbone/Wii U will sell ~280m units combined.
Well, there will obviously be a contraction of the console market just on the fact that most of the 100 million Wii owners just up and left leaving the Wii U in its current DOA state. Whether this will affect that other 180 million or so which support the PlayStation and Xbox brands is the bigger question. I don't think it will, though Xbox One will lose market share while PlayStation will gain it. However, that 180 million that PlayStation and Xbox have has shown that the audience for traditional consoles hasn't grown beyond what the PS2 and OG Xbox managed back in the day, so it certainly isn't a growth market. It is possible that PlayStation and Xbox have achieved a sort of Coke/Pepsi duopoly at this point, namely two core gaming brands that are basically interchangeable for the majority of core gamers.
It appears we agree there will likely be market contraction, then. In turn I agree the Wii is primarily the cause, but I don't understand why people are listing this as a good thing, or something which can be dismissed. The Wii was by a wide margin the most profitable home console of last generation. Saying "well it's only contracting because all the profitable consumers are leaving" is pretty good cause for concern.
I think the better question is if something like the Fire TV will appeal to the MIA Wii audience, and if Google's, Apple's, and Samsung's eventual answer to it will cause that market to be split four ways. Systems like these likely won't appeal to the majority of the PlayStation/Xbox fanbase, but appealing to the former Wii fanbase is possible. Remember that the Wii and DS were pretty much the only platforms of their era that appealed to the casual market, while we now how Google, Apple, Samsung, and Amazon as the main companies trying to appeal to that market segment now.
At the very least, I'll give Kindle TV credit for being a better device for gaming than the Ouya and a decent enough multimedia device when compared to Roku and Apple TV. I would prefer a stock Android solution for $99 though and will wait on hearing if Android Apps can be sideloaded onto Kindle TV, as the majority of Android games I own are from the Humble Bundle.