Thanks for the clarification. That does sound like it would solve the issue.
I was expecting the ability to have people request accounts that are then approved by administrator. That would have reduced the administrative headache of getting everyone's contact info ahead of time and creating accounts for them. Nonetheless, this is better than allowing just anyone to request an account! Hopefully this solves the issue, though I am still curious and unsure why Randall's suggestion was not working.
Not true. After disabling anonymous access through Portal -> My Organization -> Edit Settings -> Security everything needs authorization. Even Home page requires login.