The visibility for every event was/is set to public.
Here is what is happening:
Someone enters event information and clicks “Add Event.” The event is now visible in WordPress, but not visible on the website.
If someone (the editor or the admin) goes back into that event and hits “Update Event,” the event shows up on the website.
But, if I am logged into WP as an Admin, and open another window to the live site, I see the event that was added without the “Update Event” step ever being performed. As soon as I log off and refresh, however, the event disappears from the website, although it is still visible in WP.