Amazon built the Fire stick to be compatible with Chromecast. Google then changed the Chromecast protocols repeatedly to prevent this.
Only after that did Amazon ban the Chromecast devices from their store.
Tell the whole story, at least.
That player interface page was very device specific (e.g. the current youtube page doesn't even work on Gen1 chromecasts with 50-60Hz videos), so Amazon couldn't use those, and only supported the direct links.
Which is not that unusual — youtube could just serve a dash manifest link that directly concatenates ad and video, or build their player page better.
Google actually did it to lock people in to their Chromecast implementation (a lot of third party implementations showed up in the early days), especially for fear of DRM being broken.
We could have had an entire open ecosystem with a single standard replacing mpd, sonos etc working with everything, were it not for those actions Google took (so as a consumer and open source dev, I'm naturally pissed).
I don't believe this is true. Do you have a source?