Why not offer the beta via F-Droid as pre-release like e.g. K-9 Mail does? That could proof useful to increase number of testers and improve feedback during beta periods.
The most interesting thing during the beta are the automatic crash report analyses that Google gives us. We don’t have that on F-Droid.
Hmm or implement a mechanism, that does not require Google interaction. Fedilab solves this by offering automatic mail generation for crash reports.
Google’s method is enough and works fine. No need to spend our time developing something else. From my experience, most people “abuse” the beta to get earlier updates, instead of actually contributing something back and spending time on good bug reports. I doubt that there would be any crashes from F-Droid that we don’t get through Google Play.