How to hook watchdog script for existence of a WP page?

An easy way to do this would be to create a simple Must Use Plugin and check for your page in a function hooked to an action, like wp_loaded

function wpd_check_page_exists(){
    $page = get_page_by_title( 'The Page' );
    if( !$page ){
        wp_mail( '[email protected]', 'a message from your site', 'the page is gone!' );
    }
}
add_action( 'wp_loaded', 'wpd_check_page_exists' );

You may want to use a different function to query the page, depending on how you identify whether it exists or not, like get_post if you want to pass an ID, or get_page_by_path, or WP_Query.