Display email address instead of the user_login when user is already logged in

email-address-login.php

<?php

add_filter( 'wp_login_errors', function( $errors ){
	$GLOBALS['hb_backup_user_login'] = null;
	if ( ! empty( $GLOBALS['user_login'] ) && empty( $_POST['log'] ) ) {
		$GLOBALS['hb_backup_user_login'] = $GLOBALS['user_login'];
		$GLOBALS['user_login'] = wp_get_current_user()->user_email;
	}
	return $errors;
});
add_action( 'login_form', function(){
	if ( ! is_null( $GLOBALS['hb_backup_user_login'] ) ) {
		$GLOBALS['user_login'] = $GLOBALS['hb_backup_user_login'];
		$GLOBALS['hb_backup_user_login'] = null;
	}
});