Google account without mail on iPhone
This may be old news to many but I just hit it for the first time.
iPhone, at least at iOS 10, will not allow you to add a Google account that doesn’t have GMail associated with it. It lets you go through the whole Add an Account, authenticates, then throws the account away before you have a chance to turn off Mail. There seem to be a lot of folks posting on the Apple forums with the same issue. The solution below is also there but you need to dig for it.
The workaround: For Calendar go to Settings > Calendar > Accounts > Add Account > Other > Add CalDAV Account. Use google.com for the server name. For Contacts it’s the same except you’re adding a CardDAV account.
Why do they make you jump through hoops like this instead of using the standard Google account setup? We may never know.