You don't need a PMS for Email Only, I was only asking for you to perform a test, but given you have no PMS there's no reason to test it. Please check the spam folder of the users email account that should've received the notification.

To test the notification first ensure the group owner for example has their notifications set to receive a notification on new user join. Once done register and login with a new user then attempt to join the group. Once joined successfully check the group owners email account.

Do invite emails send out? For example try inviting an email address that isn't registered on the site yet and see if it's received. This will confirm if GJ notification function is working at all or not for your site.

An owner won't get pending approval emails for an invite. Being invited bypasses the authorization and immediately makes them approved. The only way to get an approval notification is if your group is approval type and someone clicks the Join button. If you want notifications for every join then you need to select that type of notification.