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> On completion, it is necessary to remove the .htaccess file (not a huge problem, as this can be automated) to access the site.

Or you can tell UNiTE to use a .zip file with a custom .htaccess file which works fine on the restored site.

> The issue is that on accessing the site, we are taken into the ‘Angie’ screen, rather than directly to the restored site.

I am having a double "what the heck" moment here. The two reasons are:
1. Setting "Embedded installer" to "None" (there is no "Off" option so I assume this is what you mean) does not put ANGIE code in the backup archive.
2. Your backup archive does have an "installation" directory since the SQL files are stored in installation/sql. However, UNiTE's clean-up stage removes the installation folder. This means that Joomla! doesn't see that directory and doesn't perform the redirection. And since the folder's gone there's no way it runs ANGIE, considering that the code is not there.

I suspect a code cache. Do you have an installation directory left over in your restored site? If not then you have set up a code cache in PHP. PHP 5.5 and later do come with opcache (formerly: Zend OpCode Cache) enabled by default. If this is the case you need to restart the web server daemon or PHP-FPM upon restoring the site, depending on how your server is set up.

If you do see the installation directory it means that UNiTE didn't have adequate access to remove it. In this case I would guess that you're using UNiTE's FTP mode to let it write the extracted files. Does your FTP user have the privileges to delete, not just create, files and folders?

Version 3.7 was never supported on Joomla! 3.3. In fact I can guarantee you that even if you manage to install it you will not be able to get it to run properly. And even if you manage to use it the restoration will have problems. 3.7 was written about 2 years before Joomla! 3.3 even existed.

Akeeba Backup 3.7 predates Joomla! 3.3. Moreover Akeeba Backup 3.7 is very, very old and has several known issues. Two of them are security issues we fixed last summer.

I do understand why AB 3.7 runs on Joomla! 3.3: Joomla! 3.3 later comes with a relatively up-to-date version of our FOF framework. This means that the interface can run. The backup generated has an installer never designed for Joomla! 3.3, though. If you try to restore your site you WILL lose some configuration.php settings. Not to mention the known issues, including the two security issues, in your ancient and completely unsupported version of Akeeba Backup.

Akeeba Backup 4.2.1 is compatible with Joomla! 3.3 and the only version of Akeeba Backup we will support on Joomla! 3.3.

I already told you what to check but I repeatedly got no feedback about it. I'm sorry, I have no option but to close this ticket :(