To sort of answer your question, I use Fastmail to host my email and contacts, and I self-host Nextcloud, and that has my calendar and tasks. I use Caldav to have Fastmail see my calendar/tasks, and it is entirely transparent to me.

I can also have Fastmail host my calendar/tasks, and that can use caldav as well.

There's a calendar in Kolab: https://kolabnow.com/feature/calendar

NextCloud has CalDAV and CardDAV extensions for contact & calendar syncing just like Google. It's self-hosted, though.

I use mailbox.org, they use a customized Open-Xchange. Not sure if it is the open source community edition, though.

I use mailbox too. It is privacy minded email/caldav host with great price and nice ui.

I am not sure if they use any proprietary software though.