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 extracts backup archives. Moreover, it "cleans up" after restoration by removing itself, the restoration script (which lives in the installation directory extracted from your backup archive), and the backup archive itself to ensure your site's security.
The actual site restoration –restoring the database contents, reconfiguring your site, replacing database data (WordPress backups only)– is handled by the restoration script. The restoration script IS NOT part of Kickstart itself. The restoration script is placed inside your backup archive when you are taking a backup. It is extracted into the installation directory.
As a result, Kickstart is NOT technically necessary to restore an Akeeba Backup archive. If you have another way to extract your backup archive file(s), you don't need Kickstart. For example, if you took a backup using the ZIP archive format you don't need to extract it with Kickstart; you can use any ZIP archive extraction method you prefer, such as the one provided by your hosting control panel's file manager.