Kickstart IS NOT the restoration script. You DO NOT need it to restore backups.
Kickstart is an interactive archive extraction script and a clean-up tool. It is not a site restoration utility in itself. Kickstart performs the steps required before and after site restoration, whereas the site restoration procedure itself is carried out by the restoration script which was put inside your backup archive at backup time. To make it crystal clear, here is the flow of a Kickstart-powered site restoration procedure:
Uploading files. This is done manually. You upload Kickstart and the backup archive (ZIP or JPA format) to the target site's root.
Archive extraction . This is done by Kickstart. The backup archive is extracted. At this point your site is not yet ready to work.
Restoration process. This is done by the restoration script which was put in the archive at backup time and is now extracted on your server. You are asked some questions, the database data is restored and the new site configuration file is written on the disk. At this point, your site is pretty much ready to workin most cases. The exception is in case you need to manually reconfigure your .htaccess file and/or any software installed on your site (e.g. Joomla! components, WordPress plugins, Drupal modules and so on) which stores absolute paths or URLs in their own configuration.
Cleanup of unnecessary files. This is done by Kickstart. The backup archive, the installation directory and Kickstart itself are no longer needed and therefore deleted.
As you can see, Kickstart is a very generic tool, not strictly limited to restoring Akeeba Backup / Akeeba Solo backup archives. In fact it can be used to install popular CMS such as Joomla! and WordPress. It even lets you download their installation ZIP archives directly from Kickstart itself!
The correct way to see Kickstart is as a generic, interactive, web-based archive extraction script which can also clean up after itself.