You really don't seem to want to acknowledge the answer to your questions, but just for sport I'll offer them up anyway.
As with anything in business, it's a numbers game. Putting down your ATV remote to get another device in order to make an Amazon purchase or rental is in fact a big deal. A huge reason for Amazon providing Prime video content is to get their customers proximate to premium content for rent or purchase. That's how loss-leaders work. O'Charlie's gives away pie on Wednesdays to get customers in the door on their traditionally slowest day, ordering entrees and drinks. If they did the math and found that they were spending more on pie than they were netting on increased Wednesday sales, there would be no more free pie.
If Amazon focus-grouped their video app to look at viewer conversions from free Prime content to paid purchases and rentals, their highest conversion rate would be on devices where rentals and purchases can be made within the same app. For devices like iPads and Xboxes where you have to switch to a browser to make amazon purchases and rentals, you would see some dropoff, but still see decent conversion numbers, because users are already interacting with the device. On the Amazon video app on iOS devices, if you search up paid content, it lets you tag the content and there's a "how do I watch this content?" link that explains about switching to the browser to make the transaction. I would be truly surprised if their Xbox app didn't have exactly the same thing.
For scenarios like the ATV, however, Amazon would see a huge drop-off in sales and rental conversions, because the user has to take their eyes off the screen, put down the remote, go to a computer or pull out a phone, turn it on, log in and whatever, make their transactions, put the computer or phone back down, pick up the remote and go back to the TV. Each additional step is a point where some number of users will disengage, get distracted or do something else, and with each additional step, the potential for revenue drops closer to and then below Amazon's costs for delivering 'free' Prime content.
This is exacerbated when the user has the option of skipping all those steps and simply using the remote already in their hand to switch from free Prime content in an Amazon app to paid premium content in iTunes, so the dropoff in Amazon's conversions is even greater.
So yeah, the browser on Xbox is quite probably terrible, but at least it's there, and potential digital content sales numbers for Xbox users is probably minuscule anyway, so the potential down-side for Amazon on the Xbox is also probably quite negligible. Not so with the Apple TV. Amazon is not going to use its loss-leader to drive customers to Apple, and Apple is not going to forgo its traditional cut of in-app sales just to also lose a percentage of their iTunes content revenue. On the ATV, because there's no browser, it's one of those options or the other, an it just isn't going to happen.