As a consumer, a Prime member and an Amazon Visa member who converts rewards into Amazon dollars (that usually go towards fueling my gaming habit), I'm very excited about the possible synergies if Amazon becomes involved. It would be a net positive since Amazon has a MUCH better track record of providing good customer service. The entry costs could also be streamlined for consumers through various initiatives, plus game streaming would fit in nicely with Prime as a service.
As an armchair analyst, I don't think that it makes any sense whatsoever. Hardware exists purely as a vessel for Amazon to sell their goods and services and a niche, high end gaming/entertainment box that sits in one place just doesn't seem to fit the profile with what Amazon has historically tried to do.
Plus, I seriously doubt that they have the appetite for the intense R&D and startup costs that go with the territory. I think they are MUCH more likely to go in a completely different, much less risky direction for their entry into the market.
What I COULD see happening is some kind of co-marketing and co-branding effort to better sell Xbox games and services through Amazon with a much more integrated solution than what exists today. Of course, Sony and Nintendo could do the same thing. And we'd all benefit.