My site does not take over all the block regions...

I really can't answer that, as that feature has only been tested with default themes and the Gutenberg theme.

Removing the block regions seems to be an undocumented feature of Drupal. I stole the practice from Panels module.

You could, alternatively, create a page-mysite.tpl.php file that removes calls to the regions you don't want. That template will conditionally load when the path is example.com/mysite*

OK, it is not undocumented. It is part of PHPTemplate. http://api.drupal.org/api/function/phptemplate_page/5.

It seems that the function only clears the $sidebar_left and $sidebar_right regions. So this _might_ be a bug in PHPTemplate.

If you are using a non-PHPTemplate theme, it may not work at all.

Another option would be to wrap your page regions in:

  if (arg(0) != 'mysite') {
    print $my_region;
  }

This is a limitation of PHPTemplate, not MySite. Added to the documentation.