>I'm a Prime subscriber but can't watch Prime videos (TV and movies) on my regular Android device, although if I had an Amazon Kindle device I could watch them there.

You've been able to do this for at least 3 years. Not one-step-you're-done-easy, but officially supported with no magic incantations required: https://www.theverge.com/2014/9/9/6126165/amazon-prime-insta...