Updating without Composer I've previously used Drush to update my Drupal-installations. They are installed without using Composer, and currently on version 8.7.3. However, when I try to use Drush (drush v8.1.15, using drush up) to install the currently newest version the update fails when untarring the new core. When googling I see conflicting answers about what to do - among suggestions are Drush9, updating using composer (which I tested - does not work since the page is not installed using Composer) and others. What would be the simplest, least likely way to destroy the page while being able to update (with regards to security updates)?
The latest releases of Drupal 8 are "supported but not recommended" on Drush 8. Drush 8 is still necessary if you are not using Composer to build your site. Composer is recommended for all Drupal 8 sites because sooner or later you are likely to need a module that requires Composer. If your site does not yet need such a module, though, then it is possible to continue to use Drush 8 and the pm-update command.

A few months back drupal.org changed all of the URLs used to download Drupal images; in order to continue to use Drush 8, you need the latest version, which has been updated to use the new URL scheme. Please update to Drush 8.3.0, and you should be fine.