Here’s something related, partly also about getting push mode to work …
If you don’t have a Google account, you might want to consider using Fairphone Open OS (based on Fairphone OS, but rooted and stripped of Google services, officially supported by Fairphone) or LineageOS (based on Android 7.1 already, supported by the community), which in general give you more control over what’s on your phone and what’s not.
If you so choose, you could then install the Open GApps pico package for the Google Play things (if an update is needed, you would just download the current package again and install it), or you could try to emulate Google services with free software via microG.
In any case, if interested, have a look at respective forum topics first to get an impression of what to expect.