Rather interesting article given that a crap load of other articles found elsewhere on the net pretty much lays the "blame" squarely in Apple's lap. At best, maybe Apple had a change of heart but I too doubt that it was merely an oversight on Amazon's part to not create the app. Both Amazon and Apple are two heavy competitors in this venue of streaming rentals and purchases. I don't doubt that Amazon too has been a "bad boy" where the customer is concerned in its reaction and not providing what is needed to get Amazon Prime on some other devices out there.
The only articles laying blame on Apple's lap were all from anti-Apple fanboys and completely based on speculation. This and the previous sourced articles all point to the ball being in Amazon's court.
Apple's app store is very very open. Really. Download Xcode, make something, and submit it. If a company like Amazon was being blocked from putting their app on the tvOS store, they would start making a stink about it (as Google did when it thought its apps were unfairly delayed a few years back, for instance). There is no rational reason for Amazon to keep quiet about Apple blocking them from publishing their app. And yet, they have not complained about this at all. Ergo, Amazon is highly unlikely to have been blocked from the tvOS app store.
On the opposite side, Apple has every reason to be cautious in chiding others who have not yet created apps. Make a stink about Amazon not putting its app on your platform and you make your platform look weak and undesirable. From Apple's perspective this is to be handled in smoky back rooms, not in the press. Which isn't to say Apple is negotiating from a point of weakness: Apple's juggernaut will leave Amazon's service in the dust if Amazon doesn't get on board. But that strength is a company-to-company negotiating strength, not a "popular opinion" negotiating strength.
From Amazon's perspective, I don't think their hesitation has actually cost them much other than consumer good will. Amazon's forte is not in the "collector" realm (who would much rather be holding physical media or in a pinch downloading content rather than streaming on every viewing; Amazon sometimes allows downloads, but not always and certainly doesn't make it easy). Amazon streaming (1) sells Prime memberships, and (2) makes a modest revenue stream in the form of rentals. While the former is going to take a hit, it is a yearly renewal and so won't be affected by a several-months delay in usefulness. On the latter, consumer loyalty there is completely fungible, and when Amazon comes up as one of the half dozen places to rent a movie from, they will be in no worse situation than they would have been if they'd supported tvOS day one. That said, the Prime business is far and away more significant to Amazon than single-shot rentals, both because it generates much more profit and because it builds loyalty.
What would hurt Amazon is if, come next Fall, they were still not supported on the Apple TV. Then people who own AppleTV boxes will start looking at their Prime memberships and saying that the Prime Video "benefit" is kind of useless or too inconvenient to be worth it, and some portion of them will then cancel their memberships. Most Prime memberships renew in the peak buying seasons (especially later December), for obvious reasons. That is where the decision point will be. This season I suspect won't see a massive fall-off in memberships, reasoning that Amazon is just having a hard time getting their app together and into the store; next season it would be clear that Amazon has decided to not support the AppleTV at all, and those memberships would fall off.
So, Amazon's initial play this season was to see how viable a head-in-the-sand strategy would be. If we pretend Apple TV doesn't exist, will we sell more FireTV boxes, and make more off those than we project to lose in recurring Prime subscriptions? Will those FireTV box sales themselves increase Prime memberships more than AppleTV-based Prime membership drop-offs?
I think that if we see an Amazon Video app on tvOS before Christmas it will be quite clear how that experiment went.
On a more personal note: Amazon has lost my family's loyalty. We still have a Prime membership (ours renews in June, so we are off-cycle with the majority), but come renewal time the main question will be if shipping costs are worth the membership cost. I think that this next year they might still be, but it is going to be much closer than it ever has been before. Our experience with the FireTV Stick (horrible horrible POS) has kept us from using Amazon Prime video for anything significant, and not supporting AppleTV keeps us from giving them a second shot. And as others have noted, the Prime Shipping has gotten less and less reliable over the past year or so (and a $5 credit or extra month of Prime each incidence is great but doesn't assuage the damage caused by something arriving two days after we needed to have it). We've also been hit by the false-description knockoffs on the Amazon "Marketplace" one too many times and refuse to use new merchants there (again, we can send the counterfeit goods back and get a refund, but convenience costs us). There's just too much rotten in Amazon to treat it as a no-thought-required first place to look for goods.