Even if live-streaming video games (or other digital real-time content like screenshares) was a hot market, Amazon doesn't have as good a record as Google in integrating it's acquisitions.
Look at Audible - they still have their own crufty website (which uses the same credentials as your Amazon site, sure) and discovery and reviews are less usable there than on Amazon.com.
Also see Goodreads, DPR and IMDB - all places where things have essentially gone on "as before".
What good is "integrating" acquisitions? Users don't care about your org chart.
Amazon could clone Twitch fairly easily (bandwidth is the limiting factor and Amazon happens to have gobs of it), but they still bought the Twitch brand. It would be a ridiculous waste to throw that in the trash.
I think the integration is pretty good, personally. I mostly listen to books these days, and it's usually Audible books purchased through the Amazon site.
Audiobooks (both Audible and physical CDs) are listed side-by-side other types of book media on Amazon.com. To pick a random example, if you look up Steel World by BV Larson: http://www.amazon.com/gp/product/B00FCXPC94/ - you'll see the paperback, Kindle Edition, Audible audio edition, and sometimes others.
Some titles have "Whispersync for Voice", which I think is an impressive type of integration. With those titles, you can sync your progress between the Audible and Kindle formats, so that you don't lose progress when switching from reading to listening. Additionally, when you own one format, the price of the other format drops substantially (if I recall, usually to a few dollars). I count that as a deep and sophisticated type of integration between the Kindle store and Audible.
Whether I visit the Amazon.com site or Audible depends on my motivation. If I want to find a good book, and I'm willing to either read or listen, then I'll hit Amazon. If I'm looking specifically for an audiobook, I'll hit Audible. Also, I think the Audible site might have been redone since you've last seen it. The home page doesn't look like what was around before the acquisition. I see a modern, Amazon-style product grid on audible.com when logged in. It's also integrating content from Amazon such as "Based on your Amazon and Kindle book purchases, we think you'll enjoy...". The Audible site to me looks like a curated window into Amazon.com now.
Look at Audible - they still have their own crufty website (which uses the same credentials as your Amazon site, sure) and discovery and reviews are less usable there than on Amazon.com.
Also see Goodreads, DPR and IMDB - all places where things have essentially gone on "as before".