/**
 * Disables the WordPress API for unauthorized users.
 */
function disable_wp_api() {
	add_filter( 'rest_authentication_errors', function( $access ){
		if ( ! is_user_logged_in() ) {
			$access = new WP_Error( 'rest_not_logged_in', 'You are not currently logged in.', [ 'status' => 401 ] );
		}

		return $access;
	} );

	// also remove actions added by wp-includes/default-filters.php
	remove_action( 'wp_head', 'rest_output_link_wp_head' );
	remove_action( 'wp_head', 'wp_oembed_add_discovery_links' );
	remove_action( 'template_redirect', 'rest_output_link_header' );
}

disable_wp_api();