This incident affecting the owner of a popular site who used “site builder” software from his hosting company is a good example of the downsides of this solution – you can’t (easily) upgrade the site, archive it or move it to another host.
If you use a well-supported content management system like WordPress or Drupal there are tools to help you upgrade and migrate. We recently upgraded another site that had stopped working because it used a mix of old systems (Joomla, Magento, phpBB, CiviCRM, WordPress) some of which no longer worked properly in Ubuntu 16.04 LTS. We moved it all to WordPress without losing content, using standard plugins. Keeping it updated or moving it to a new host is no longer a problem.