For years I worked in the "shareware" industry and this worked extremely well (this was from 2000-2011). We didn't sell games, but rather time-limited software. In our case, we let the user have all of the features, but it only worked for a month. After the month, you either needed to uninstall the software or pay for it.
When I ventured off to do my own Play Store software I did what you suggested. I had a "lite" version and the full version. The tricky part was finding a way to migrate data between the two versions without using a server in the middle.
While this doesn't apply to games, it can still be done using in-app purchases. Supply a feature- or time-locked version, allowing the user to purchase the full app if they want more power.
Now, if only we could find a way to properly implement upgrade pricing...
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