Pros and cons of each?
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Google play has auto update, whereas f droid has semi automatic updates where it takes more effort.
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The Google play flavor comes with support for FCM (i.e. push notifications via Google infrastructure). The f-droid flavor relies on regular polling for events (so that messages might come in a little bit later, though I am not sure whether it can keep a connection open for real time push notifications).
Another solution can be use Aurora Store, you download always from google play store but using anon account create from Aurora so the request dont come from your account, but sometimes can appen that you have package in conflict but it’s rare.