I agree with you. It's so weird that they host their VPN app on F-Droid but not their email app. It seems like it wouldn't take too much effort on their part to add Proton Mail to F-Droid.
Protonmail
There is now an official Protonmail Community at [email protected]
I remember reading a discussion somowhere and it seems they won't add ProtonMail app to the F-Droid as it relies on Google Play services for push notifications.
If that's the case, it makes me wonder why they designed it this way. Their main selling point is privacy, yet they are so closely aligned with Google's services? Seems very questionable, especially since other email clients such as K-9 provide notifications just fine on Android.
This is the sort of thing that gives me second thoughts as a paid subscriber about using Proton's services at all.
Honestly I don't know, but this is the reason that I passed on the Proton when I wanted go for paid tier and I decided to go with Tutanota instead, as they have all components of their apps open source, as far as I am aware.
There's the one con that they don't allow to use any 3rd party email apps, but from what I have heard, ProtonMail Bridge is also not that great of a solution.
On android there is a tradeoff, either you use googles notification services, or running your own, which will cause battery to die faster. They decided it'n not worth it, and they also don't want to release it without notifications at all, as it seems.
Weird since they apparently thought it was worth it for their VPN app, but not their email client.
VPN app doesn't have to poll server for incoming messages. And, anyway, it is expected to drain your battery when connected
I overall trust APKs from IzzySoft as those are just fetched official releases compiled by app developers themselves. You can verify that yourself by downloading older version APK from ProtonMail GitHub repo then updating the app through F-Droid. If it updates then it means that the APK from GitHub and APK from F-Droid are signed with the same key.
Another way would be to use something like Obtainium to track new releases on the official ProtonMail repo.
If in doubt, hit up their GitHub
https://github.com/ProtonMail/proton-mail-android
And use Obtainium to get updates.