So far I came up with a trick that takes advantage of the BETA channel of Google Play: every device uses the same Google account which is registered as a beta tester for my private app (and by private, I mean I don't want it to be available for the rest of the world).
When the "beta" account is set on the device, I go through Google Play and install the app as usual. Then, provided that GPlay is set to update every application automatically (enabled by default), any new version of my app pushed on the BETA channel ends up being updated on every device that shares the so called "beta" Google Account. Looking good.
However this little trick does only work as long as I don't encounter the following error message in the notification bar:
Account action required
which triggers every now and then and spreads over almost all my devices after a couple of days. This error message forces you to re-enter the Google account credentials, which is quite a big problem considering that the end users of the devices do not and should not know those credentials.
In short: why does the error message appear and how do I get rid of it?
EDIT: I am open to a solution that does not use Google Play but another app store such as the one from Amazon. Yes, I do realize this additional remark is borderline for a Stack Overflow question.