Can I make a Customizer dropdown-pages list include private pages?

Unfortunately there doesn’t seem any way to hook into the underlying get_pages call that the control dropdown-pages uses, so you’ll have to build your own set of choices (that includes private pages) and use the standard select control type instead:

if ( ! class_exists( 'WPSE_406789_Walker_Page_Options' ) ) {
    class WPSE_406789_Walker_Page_Options extends Walker_Page {
        public $choices = [];

        public function start_el( &$output, $data_object, $depth = 0, $args = [], $current_object_id = 0 ) {
            $pad = str_repeat( ' ', $depth * 3 );

            $this->choices[ $data_object->ID ] = $pad . get_the_title( $data_object );
        }
    }
}

$walker = new WPSE_406789_Walker_Page_Options();

$pages = get_pages([
    'post_status' => [
        'publish',
        'private',
    ],
]);

$walker->walk( $pages, 0, [], 0 );

$wp_customize->add_control( 'resources_page', [
    'type' => 'select',
    'choices' => $walker->choices,
    'section' => 'key_page_locations',
    'label' => 'Resources page',
    'description' => 'A page for private resources',
]);

I’ve used a custom walker here so that you still get the   padding that the default wp_dropdown_pages generates to visualise the page hierarchy.