Amazon Instant Video is the only advantage a Kindle Fire has over a Nexus 7. Without that, there's absolutely no reason to buy a Kindle Fire.
Because on a Nexus 7, you might also go to the Google Play store to buy books, music, or videos. But Amazon has removed that from the Kindle Fire.
That is the madness of which I speak. Because they are afraid that if I don't use the Kindle Fire I won't buy videos, they therefore won't let me buy videos unless I use the Kindle Fire. It seems like a self-fulfilling prophecy to me. That same logic would call for them to not have a Kindle app on the Nexus 7, would it not?
Like I said before, the Kindle Fire gets a lot of exposure due to its prominent position on Amazon's home page. This makes it much more likely that they'll get a lot of sales from consumers who don't really do much research. This is especially true during the holiday season - you don't do nearly as much research when buying something for somebody else as you do when buying something for yourself.
Even then, like you said, there's a certain lack of logicality in Amazon's decisions. Then again, they haven't been in the mobile device market (as an OEM) for too long, so it's not entirely impossible that some MBA came up with this harebrained scheme for driving sales of content from Amazon's digital stores and got it past Bezos.
I imagine they took some base stable version of android, and built their services platform on top of that. They have their own ridiculously huge Web Services platform that they want to provide that at a system service level to application developers.
To do this quickly, they had to throw out a lot of hardware support both at the system level for things like gyros, hardware buttons, GPS, etc (all the things the iPad and Google Tablets have), to be able to undercut the manufacturing price to an attractive retail market value.
There is a good chance their video streaming leverages system services built off an increasingly "Amazon" branch of early stripped down Android code.
Once they have their platform, they can focus on rolling back in the hardware and services to support a larger eco-system.
Ultimately, I think they are trying to enable a hub to provide all of their web technology through a mobile device. They've chosen android to safe themselves a lot of time plus they aren't OS guys.
Now they are the awkward cousin that doesn't have a fully realized value proposition and is semi-cross functional with other devices and more enhanced platform.
In fact, developing for Kindle Fire would probably feel a lot like porting an Android app (that was originally written for iOS in Objective C) and designing around the current hardware peripheral limitations (and the fact that "Google" intents would be completely ignored).
What they could be betting on is the shear technical force of their infrastructure and web services. If they can bring high-quality content to their users consistently, people will feel the comfort of Amazon more than other eco-systems.
This puts Amazon in a weird position because they are competing with Google more than Apple, really. Especially since they chose Android. Google expected making and distributing Android would increase the gravity to their platform, but its totally possible they understimated Jeff Bezos and his ability to leverage it.
That said, the horrible app store and out of date android build get old. More than half of my apps on it are APKs I got from the support staff of the various companies and have to manually update. Since the Google Play store also won't let you just download the APK or run on the Kindle Fire (two play at that exclusivity game!).