ironically ios has this built in and one can sync contacts/calendar from iphone to $selfhosted without even touching the app-store. not so for android
Yeah, I bought the DAVx/ICSx pair (as well as CalDAV-Sync and CardDAV-Sync pair) to get Android syncing. I found it more understandable than when configuring Fastmail on iOS given the way Fastmail makes you (or made when I initially set it up years ago) use separate app-specific passwords.
You can install DAVx5 from F-Droid not touch the Play Store.