Are there any easier ways to update a site than following the official instructions?
If you have access to ssh in your server install drush (drupal shell) http://docs.drush.org/en/master/install/ 

This will only need to be done once and and then it will be easy to apply drupal core updates from the command line

drush up drupal


or any other module updates

drush up views


But I would recommend applying ALL security patches at once (core and modules security updates) using

drush upc --security-only


all the above commands will do all the necessary actions as described in the instructions that you linked above and in case of a bad update the updates are reversed with error messages displayed.

To learn drush read https://www.ostraining.com/blog/drupal/drush/ but the installation instructions might be a little outdated. You can also learn more drush commands at drushcommands.com