Kickstart only extracts the backup archive. It has nothing to do with the actual restoration. The restoration is performed by the Akeeba Backup Restoration Script which is included in the backup archive at backup time. When you extract your backup archive this is extracted in the folder /installation.
If clicking the Run the Installer button does not work for you it is always possible to access the /installation/index.php URL on your site to run the restoration script. After all, all that button does is ask your browser to open a new tab with that URL.
Kindly that the code which open a new browser tab has not been changed in Kickstart for many years now. The only change in Kickstart 7.1.0 is what happens when you click the Clean Up button. It will now also clean up PHP's OPcache. The last time we made any change affecting the interface of Kickstart was in 7.0.3.
Finally, I'd like to point out that you claim to be using Akeeba Backup 7.0.5. However, there is no such version. The version numbers went from 7.0.2 (released February 11th, 2020) to 7.1.0 (released March 20th, 2020). As a result I have no idea which version of Akeeba Backup you are using and whether it is compatible with PHP 7.4. Compatibility with PHP 7.4 was added in Akeeba Backup 7.0.0.b1 released December 2nd, 2019.
Nicholas K. Dionysopoulos
Lead Developer and Director
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