Enable client to add users to user group but not set permissions
doesn't work if the client doesn't have the "Assign user groups and permissions" checked.


That's not a limitation of the plugin, but a requirement of Craft.


  Is there a way to force every new user created in the CP by the client into a default group?


Yes, that's what the previously linked plugin will do, with the caveat that the currently logged in user has to have assignUserPermissions permissions.

The only way I can think of to do what you're trying to accomplish is build the user administration stuff for your client on the front-end of the site.  That way - though a plugin - you've got complete control over the user management business logic.