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You are using FTP instead of SFTP. Absolute paths usually don't work on FTP servers as their sessions are "jailed" to a specific directory, usually your home directory. Instead of an absolute filesystem path you would need to enter a path relative your FTP initial directory.

Or, even better, just use SFTP without changing anything else. 

SFTP is encrypted for both authentication and file transfer, FTP is neither. SFTP uses the native Operating System permissions management system, FTP is a hodgepodge of technologies with questionable security results at best. SFTP lets you use absolute filesystem paths, FTP has that stupid notion of the "initial directory" and jailing sessions in a subdirectory which leads to a big headache trying to figure out what is pointing where.

When in doubt, SFTP.

Hm, if you can't find the server could that mean that you don't have SSH (upon which SFTP runs — it's file transfer over SSH) set up on the target server, or don't have access to it? I think it's worth investigating this with your host because it's a far more secure, faster file transfer method than the 50+ year old FTP protocol (not even exaggerating; FTP was formalised on April 16th, 1971).

Another hint here is that you may need to use a different method. Remember that there are two variations each of FTP and SFTP:

• FTP, using PHP's FTP methods which should be available on all servers.
• FTP over cURL, using PHP's cURL integration extension. This extension is usually enabled and the cURL library is usually compiled with FTP support. I have seen very few live servers not providing this.
• SFTP, using PHP's optional SSH2 extension, if available and enabled (ask your host, please).
• SFTP over cURL, using PHP's cURL integration extension. I have seen many live servers which do have the PHP cURL extension installed, but the cURL library they are compiled against doesn't offer SFTP support.

So, your mileage may vary.

Do keep in mind that SFTP listens to a different port than FTP. Typically, FTP listens to port 21 and SFTP listens to port 22. Some hosts may use different ports. If unsure, ask the host.

If you have to use FTP or FTPS (FTP over explicit SSL/TLS) you will need to find the correct path to the site's root. Remember, that was the problem all along. As per the documentation's suggestion, log into your FTP server with FileZilla, navigate to the site's root and copy the path appearing at the top of the right hand navigation pane. You might see that it's something like /public_html (instead of the absolute filesystem path which is more likely something resembling /home/myuser/public_html). Yeah, I understand how convoluted it is. There's a reason —or, rather, hundreds of reasons!— everyone in their right mind hates plain, old FTP with a passion.

For what it's worth, the Joomla version is irrelevant. Both versions of Akeeba Backup (8 for J3, and 9 for J4) use the exact same backup engine which provides the FTP and SFTP connection facilities. The interface is also identical, just adapting the HTML output (basically, CSS classes) for Bootstrap 2 in Joomla 3 and Bootstrap 5 in Joomla 4. I have tested both against the same target servers, both local development servers and live testing servers. Both do connect just fine to the same servers and can transfer the sites using the Site Transfer Wizard. I would say the difference is just the source or target servers you are using with each CMS version.

Oof, that's a weird server setup. In the hosts I have used add-on domains are hosted as subfolders of the same user account. SFTP connects to the user account's folder and I can see all of the domains' folders. Well, if the host doesn't help what can you do?