How to desactivate jquery_update logic for the homepage
The reason why your theme_preprocess_page function isn't working, is because the jQuery update module uses a clever implementation of hook_theme_registry_alter to ensure it's implementation of hook_preprocess_page is called last.

What I recommend you do is create a new module and implement your own hook_theme_registry_alter and hook_preprocess_page functions. See a simple example below which has taken some code snippets from the jquery update module:

function mymodule_theme_registry_alter(&amp;$theme_registry) {

  if (isset($theme_registry['page'])) {
    if (count($theme_registry['page']['preprocess functions']) &gt; 0) {          
      if ($key = array_search('mymodule_preprocess_page', $theme_registry['page']['preprocess functions'])) {
        unset($theme_registry['page']['preprocess functions'][$key]);
      }
    }

    // Now tack it on at the end so it runs after everything else.
    $theme_registry['page']['preprocess functions'][] = 'mymodule_preprocess_page';
  }
}


This hook implementation tells Drupal to invoke our custom hook_preprocess_page function after all other modules have run their hooks.

Then we need to define our own custom implementation of hook_preprocess_page which will alter the scripts included on the front page:

function mymodule_preprocess_page(&amp;$vars) {

  // Only do this for pages that have JavaScript on them.
  if (!empty($vars['scripts'])) {

    $template_files = $vars['template_files'];
    $scripts = drupal_add_js();

    if (in_array('page-front', $template_files === TRUE)) {
      // Here I unset all the javascripts that I don't want to be loaded in the homepage
      unset($scripts['theme']['themes/"theme_name"/js/jquery.jcarousel.min.js']);
      // Once cleaned up all unncessary javascripts, I load them into $var['scripts']
      $vars['scripts'] = drupal_get_js('header', $scripts);
    }
  }
}


Using this approach in your own custom module will make sure that any alterations you want to make to your javascript includes will happen very last.

Good luck! :)