Apple is not going to have Amazon live streaming because the two companies are in competition with each other. If the viewer was streaming from Amazon, Amazon would be making the money and not Apple. I doubt if it will never happen that both are available at the same time on Apple TV.
The Apple TV only offers services that it has agreements with. Amazon and Apple both make products that compete for consumers attentions. Apple offers Netflix, Hulu Plus (recently added) A few sports services and iTunes. Although these services are great and offer a lot, the Apple TV doesn't offer any additional way of adding new services. If Apple doesn't have an agreement with them, they won't have their services on the Apple TV. My suggestion is to find a media streaming device that has what you want to view, purchase their device and try it out. My experience with Roku, Netgear Neo and the Blu-Ray players with wifi services are extremely laggy, clunky user interface and LONG load times. Besides offering many services. I would prefer quality over quantity.
Its a good question. I imagine it has to do with licensing agreements. Amazon or Apple may not be willing to make this happen. What it would take is for them to come to an agreement and add an Amazon app to Apple TV.
Unfortunately that will be up to Apple & Amazon to answer. I am somewhat disappointed that since the death of Mr. Jobs, Apple has been very creative about ways to get people's money but their product innovation has not kept up the same pace (for example you need a pretty new computer to airplay the screen into the apple tv besides the newest MAC OS X (Mountain Lion). I truly hope that Amazon is "developing" an app for the apple TV. When it comes to the streaming box, I do not think there is a "best". Look at your needs and what you are trying to do and your actual hardware and base your decision on those