Yes the Nest doorbell (battery) will announce visitors (i.e. anyone who pushes the doorbell button) on your Google home mini's. Make sure the setting Settings | Doorbell | Visitor Announcements is turned on.
Turning off all the other notifications is done in Settings | Events | Seen Events. There you can specify if you want notifications for people, packages, animals and vehicles. You can also use the zone feature to limit the area where notifications are triggered.
E.g. we have it set that we get notifications only for people and packages on the porch and path to the front door. The camera also catches people and vehicles on the street, but since these are outside the zone, they do not trigger a notification.
