Proton Calendar is a good choice I'd say.
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If you want something hosted with full privacy, yet importable in the standard calendar apps, then I don't think you'll find anything for the time being.
Protonmail and tutanota both provide full privacy and are hosted options, but they require their own apps since they want to ensure reliable E2EE. I suppose you either need to re-evaltuate your requirements or wait for a suiitable product to be launched.
NextCloud for the self-hosting server stuff.
DAVx5 for the syncing to Android's native calendars.
Any calendar app that uses the native Android calendars to view them. I'm using DigiCal.
as much as this is a good recommendation (god bless DAVx5), Self-Hosting isn't what OP is looking for.
Doh! I missed that. Apologies.
It sounds like EteSync is exactly what you're looking for. A hosted NextCloud instance + DAVx5 is admittedly a bit lacking in the "full privacy" department, and Proton or Tutanota doesn't integate with the standard calendars, but both would also be valid options IMHO.
Unless referring to encryption, can you expand on why NC with Dav isn't private? I've got both selfhosted (in home) and haven't picked up on any traffic sharing my calendar to third parties through my logs and alerts, so would like to know more of what you've seen to see if I have a gap in my automated alerts.
Edit: unless your answer was just in response to the OP wanting an already hosted system, then yeah I can understand what you mean there.
Will this help: https://lemmy.world/post/990528
Sorry for not being clear: I am looking for an alternative calendar hosting, not an alternative to the google calendar app.
Ich betreibe (kommerziell) einen davical Server. Da läuft sogo als web interface und mit davx5 kannst du das komplett offen und per https auf jedes android device syncen. Funktioniert seit Jahren zuverlässig und kostet nicht die Welt.
Das fällt allerdings in die Kategorie selfhosted.