Amazon’s weaknesses are in the first case high-end design And in the second enterprise.
Just design generally, including their website ... anecdotally, so many people I know don't really know that Amazon Prime Video exists, or if it does exist how to find it. Likewise for Music.
Apple has this discoverability problem too (Apple Music doesn't have a web UI, it might as well not exist if you can't install iTunes, and then Apple wonders why its social features don't take off).
For its devices, Amazon also seems to share Google's general obliviousness, e.g. they just announced an alarm clock with a camera. Who wants a camera pointed at their bed all night? Their Fire Phone pioneered this over-expensive faux-3D UI, which on paper sounds great, but anybody could have said they weren't going to get the development support they needed to really sell it, and most users aren't going to care. Design is more than just veneer, you've got to make stuff that people want... Amazon's current approach seems to be "well if we just sell it cheap enough, we'll get customers."
To be fair, there's also a (massive) market for that.
The iPhone music client is not very good. Maybe that’s more of an issue than the lack of web client.
I know these things exist, but I don't use them much even as an Amazon Prime member. Why? Because Prime Video UX is poor compared to Netflix (and Netflix is no great shakes either, but it's better than Amazon), and Music has a pretty poor catalog and again a pretty poor UX.
And this is mostly because Netflix wants to hide the true extent of their library.
Hulu just had a recent, awful re-design too. They cover in-active thumbnails with gradients, you can barely see anything.
Apple TV (which I like as among the best) also has a ridiculously unergonomic remote and very confusing UI (do I want to press "menu" or "home"? And now the TV app is 'home' by default, it's even weirder...somehow, even though I know I think how it works, I always seem to press the wrong one.) And no picture-in-picture either, though their processors are plenty fast enough to support it. But their standard built-in TVML-based interfaces are functional, if a little boring, and the UI is responsive. And hey, at least they aren't showing me ads in the UI or tracking what I watch. But still, lots of little problems.
So much room for improvement here, by everyone.
I still believe they make great products, just not as great as they have been.