Technically, you can probably ship something based on pure AOSP, but your competitors will have had pre-AOSP access to the code for multiple months and there are no guarantees that AOSP even works without Google's magic proprietary mojo. CyanogenMod had to develop the missing pieces when Google sent their C&D over the proprietary bits. I would suspect defensive things designed to make it more difficult for Amazon to leech AOSP are either present or coming soon particularly with the rumors that Amazon is working toward introducing a phone.

Amazon has forked Android, but by doing so they aren't allowed to use any of Google's apps in their devices.

What it boils down to is that Google has a final approval over any Google-logo Android device. For phone OEMs, this looks like absolute and arbitrary control by Google, but, in reality, it's a three-way negotiation between the OEM, Google, and, in the US, the carrier controlling the channel for the product, with any one of them able to sink a product before it reaches the market.

Google has never seen fit to make licensing terms for Google's ecosystem apps (Maps, Play, etc.) transparent. That's a pretty sharp contrast with Android as an open source project, and even with the way Windows Mobile (not Windows Phone) was licensed.

So Android is open, on the one (AOSP) hand, and un-transparently proprietary on the other hand. That leaves an obvious market opening for a truly open OS. But, so far, Tizen, Open WebOS, or any other possible contenders have been insufficient and/or under-developed. Maybe Canonical will get it right.