How are you browsing your Amazon content? I see search bars on the 'All Content' [1] page, and also on each individual page, like my movies and shows [2].
Though it seems like the interface is pretty rubbish in the Prime Video section [3], so maybe that's where you're looking?
[1] https://www.amazon.com/hz/mycd/digital-console/contentlist/a... [2] https://www.amazon.com/hz/mycd/digital-console/contentlist/v... [3] https://www.amazon.com/gp/video/mystuff/library
Roku based TVs and their mobile app. And the website - #3 in your list above. Never seen those other two links, and they don't show up when you're trying to watch things on your tv.