Only available on Amazon Fire TV, ROKU or AppleTV. You won't find it on Prime Video. And, the DVD release will only be on Blue Ray.
After playing around with this a bit you were right in the first place. The Packers web site says the same thing.
I found the Packer ap as you described drilling down through the Fire Stick's app store where the app is free. Amazon Prime does not have an app store by it's nature and while you can search for "Packer" content in Prime you will find some but you won't find the specific programming you mentioned. I went through a smart LG TV's ap store, which is pretty limited, and there was also nothing of the sort.
I thought about device specific content limitations years ago when I opted for a monitor-type TV and the external Fire Stick, saving a couple of hundred dollars, while having the flexibility to swap for another external device (and it's content access and performance) for cheap later on if needed. I never thought about it again until now because Netflix and Prime is all I ever watch outside cable, not being much of a TV junkie. I wouldn't even have those subscriptions if my wife didn't insist.
We tend to think about these devices, internal to a smart TV or external, in terms of their download performance, controls and user interface. They all access Prime, Netflix, Hulu and the other major names. But as this evidences, content is device specific at the margins.
