Functional testing without using simpletest
I ended up working with Behat (http://docs.behat.org/en/latest/) and Mink for Behat (http://mink.behat.org/en/latest/) to write tests that emulated a user using a browser on my website.

However, I still needed Drupal integration for quick access to nodes, users, and roles.  I found this project (https://www.drupal.org/project/drupalextension) which incorporates Behat, Mink, and the Drupal API.  This allows me to write tests like this:

@api
Scenario: Logged out user can access the custom Landowner registration page
    Given I am not logged in
        And I am on the homepage
    When I click "Register here"
        And I am on "/register"
        And I click "Register as a Landowner"
        And I am on "/register/landowner"
    Then I should see "Landowner Registration"


Now I can test a Drupal site without the issues involved with Simpletest.