Restricting Alerts for Saved Searches is a misnomer

I (administrator) created a saved search which I want to alert specific users whenever something specifically changes in the search results.

The documentation states:
Setting the audience for a saved search restricts its appearance in list and sub-list views, dashboard views, saved search e-mail alerts and preferred search forms. Only users who are members of the defined audience will see the saved search on search related lists.

However, I can't get this to work. It seems in order to make the saved search "Available as an Alert" you need to check the "Public" checkbox (all roles can view). If I then go to the Audience tab to restrict to only the people I want to be able to see this search in the first place, I uncheck the "All Roles" checkbox and select only the employees I want to have access. This automatically unchecks the "Public" checkbox AND unchecks the "Available as an Alert".