Are you using composer to manage your website dependencies? The answer to this question should be definitely yes! I work with many website designers and developers throughout the UK and it still amazes me that so many are not using composer to manage the code in their websites. I believe any web developer using open source content managed systems like Wordpress or Drupal, using composer to manage and update your sites core files and contributed modules and themes is a no brainer!
If you are not using composer yet, then do it! Using a software like Putty allows you to easily access the command line using SSH. So instead of manually uploading core files and then having to ensure that all it's dependencies are installed and in the correct places you simply run one command to update your cms, module, plugin or theme.
An example of updating Drupal core and all of it's dependencies in one go you would use:
composer update drupal/* --with-dependencies