If Amazon won't let the video play out over an HDMI cable (or via Apple TV and wifi) it is because they don't want to you do that (and thus doing so is considered stealing). I can't remember if it will play via either of these two methods because I rarely watch Prime videos at home, and when I do, I can get them on my computer screen or Tivo (well, only the ones I pay to rent...not the ones that are free for Prime members). HBO Go used to have the same restrictions, which was kinda stupid because in order to have HBO Go, you had to be an HBO subscriber. But if they let you play out over HDMI, then you could do that anywhere....which is nice. Finally they gave in and now if you have HBO on cable, you can play out anywhere. In the case of HBO, that makes more sense since subscribers pay a lot of money on cable every month. In the case of Prime, not so much since you are getting this stuff on top of upgraded shipping for $90 / year, a lot less than what the cable companies get (more so for people who order a lot of stuff from Amazon, which most Prime members do because it makes sense to do so).
If you set up a HTPC, then problem is solved. You simply go to Amazon on your HTPC and play out to your TV (which is what I do).