How to make posts under custom post type not generate a URL / post

Looking at the class and function you’re using, this method is a little problematic:

  public function buildPostArgs( $slug, $singular="Post", $plural="Posts", $args = array() )
  {
      $args = wp_parse_args($args, $this->postDefaults);

      $args['rewrite']['slug'] = $slug;

      $args['labels'] = $this->buildPostLabels($singular, $plural);

      return $args;
  }

If we ignore your problem, $args['rewrite']['slug'] is going to generate a PHP warning if you set rewrite to false for trying to use it as an array.

So if we separate out the call into a variable like this:

$args = theme_build_post_args(
    // $slug, $singular, $plural
    'integrations', 'Integration', 'Integrations',
    [
        'menu_position' => 21,
        'hierarchical' => false,
        'supports' => array('title', 'revisions'),
        'taxonomies' => array('categories'),
    ]
);
register_post_type(
    'Integrations',
    $args
);

We can now overwrite what your helper did, e.g.

$args = theme_build_post_args(
    // $slug, $singular, $plural
    'integrations', 'Integration', 'Integrations',
    [
        'menu_position' => 21,
        'hierarchical' => false,
        'supports' => array('title', 'revisions'),
        'taxonomies' => array('categories'),
    ]
);

$args['public'] = false;
... etc ...

register_post_type(
    'Integrations',
    $args
);

In particular look at what it defines in $postDefaults and override those specifically to match what you want.

You might also want to ditch the function and just use the label setup parts to simplify things:

$builder = new theme_PTTaxArgBuilder;
$args = [
    'labels' => $builder->buildPostLabels( 'Integration', 'Integrations' ),
    'menu_position' => 21,
    'hierarchical' => false,
    'supports' => array('title', 'revisions'),
    'taxonomies' => array('categories'),
    'public' => false,
    'show_ui' => true,
];

register_post_type(
    'Integrations',
    $args
);

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