You're right the scenario you're describing really does not provide good ux. However, there're reasons (and a workaround) for this.
It is important to know that in earlier versions of Android, there were only two ways of navigation:
Back (the arrow) and Home (the house)
While Home was always getting you back to the home-screen (a global escape-hatch), Back was getting you one screen back. These two rules are still true.
This caused the following problem: when you moved from screen A over B to C, it would need two Backs to return to screen A. If you think of an app with more than two screens, this becomes a serious problem.
Also, circulations were pretty common: You navigated from A to B then to C and then to B again. Hitting Back will now lead you back to C, although you might expect going back to A (if you expect it or not depends on the context).
To overcome these problems, the Up navigation was added together with the ActionBar in Android 3.0.
Up introduced an in-app escape-hatch. When you click it, you would move to the hierarchical parent of the current screen. (See http://developer.android.com/design/patterns/navigation.html for more details).
Also, when you enter an app from another app (which is a quite common pattern in Android) Back will get you back to the application you came from, while Up will get you to a different screen in the newly opened app.
The last piece of history is, that the Navigation Drawer was added after the Up-caret. I guess that google saw implementations of a drawer in more and more apps and found it pretty useful. They added it into their support-libraries so that everybody could use it.
Now, go a little further with your example:
"go onto the google play store app, click the apps category, then games, then in game offers, then a game title." Now, Scroll down and click on an app of the "similar apps" section. Repeat this multiple times. Tapping the "up"-caret will now send you to the "Games" overview page again., which is nice.
Since you can't put every possible screen of your app into the Navigation Drawer, you need other ways of navigation. But Back won't work in some cases, so you still need the Up functionality. Unlike the drawer which is provided by google as part of the support-library, Up is part of the Android platform. That's probably the reason why both is located at the same spot. There was no other "reserved" place to put it (and users new it should be there from other apps).
By the way: I don't think your use case is a quite common. I guess that users normally don't want to navigate from app details to "my apps". However, you can still access the Navigation Drawer by pulling it out from the side. There is unfortunately no way for the user to discover this feature (except trying).