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Joomla 2.5 cannot run on PHP 7. It became end of life in December 2012. PHP 7.0 was first released in December 2015 and has major differences from PHP 5, to the point that Joomla 2.5 will not run correctly, if at all. The only reason your site still runs on PHP 5.6 (even though it was also released after Joomla 2.5 became End of Life) is that PHP versions 5.5 and 5.6 did not introduce any major backwards compatibility issues, unlike versions 5.4 and 5.3 before them.

Regarding your issue, the version of Akeeba Backup you are using (4.7) has long been end of life since 5 years ago. Versions 4.7.6 and 4.7.7 were security releases but they didn't renew support for this old version. Moreover, it sounds like a permissions issue on your local server, not a PHP version issue. Even if you could overcome it, though, your site would still be unable to run on any PHP version newer than 5.6 which means that it's the end of the line for that site.

You should really have updated that site between January 2012 and 2014. The tooling for updating to Joomla 3 has really not been maintained since. Trying to find servers with PHP versions old enough to restore the site is already quite difficult since even PHP 5.6, the last of the PHP 5 versions, has become end of life on December 31st, 2018.

I'm sorry, but you probably have to either rebuild that site on a modern Joomla version or retire it.

Regard the problem, I also agree with you that seems to be a permission issue, but I can't explain myself why.

I have not used Laragon so I can't help you there. I set up my own local servers with ownership and permissions I control and understand so I don't run into problems like that :)

Since your problem seems to be with the server setup, trying to use a different version of Kickstart is unlikely to help you. You can, of course, but I don't see how it'd help.

Just another question: could you suggest me a way to backup articles and categories to recover them on a fresh installation of the latest Joomla ?

The data structure has changed substantially since Joomla 3.2 and thew introduction of tags. Now every article has an entry in the content table, an entry in the assets table and an entry in the ucm_content table. The former can mostly be transferred. The assets table doesn't translate between sites, let alone Joomla versions. The UCM content table didn't exist before Joomla 3.2 Therefore you can't do this kind of transfer using just the database.

You'd need software like SP Transfer which does exactly that, transfer content between sites. But first make sure you get your site up and running on localhost.

You can always use something like MAMP Pro (Windows, macOS) or WAMPServer (Windows) to run multiple PHP versions locally. At least with MAMP Pro you can have each site run on a different PHP version. So you could have the Joomla 2.5 site run on PHP 5.6 or earlier and the target Joomla 3.9 or 4.0 site run on PHP 7.4 which is what SiteGround is most likely using as the default now (it was 7.3 last year).