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This message means that the file libraries/vendor/autoload.php does not exist. Does the file exist? This is part of Joomla and is backed up and restored unless you explicitly exclude it, or one of its containing folders. 

Another reason this might happen is if there's a defines.php file in your site root (or you have edited includes/defines.php) setting the JPATH_LIBRARIES constant to a path which does not exist in your new site. This file is NOT changed by the restoration process. It's something that pretty much every Joomla site I have ever seen does not need to do and everyone who's not a core developer don't even know it's even an option. On a regular site, there is no defines.php in the site's root; if there is one on your site, remove it. There should only be a file includes/defines.php whose contents you can see in https://github.com/joomla/joomla-cms/blob/5.3-dev/includes/defines.php. Check the contents; if they differ to those on Joomla's official repository I linked above, change it.

The other thing you need to check is whether the media/vendor folder exists. If it doesn't exist, go back to your backup settings, make sure it's not excluded; it's a critical part of your site. It's Joomla's core JavaScript.

You have not given me enough information to tell you exactly what happened, so my response to this will be a bit longer and cover more ground than I'd ideally want to.

Akeeba Backup automatically excludes the folders which Joomla itself reports contain temporary and cache files. These auto-excluded folders have red icons. They cannot be overridden (enabled) manually. If that is an unexpected set of folders, that's always a self-inflicted problem, albeit one you might have easily missed. The temporary and log folder in Joomla are configurable in the Global Configuration page. If you eave them empty they are equivalent to the site's root. You can also override Joomla's cache folder in Global Configuration, though leaving it empty simply means "don't override" in this case. Using the wrong folder can and will cause problems. The worst case scenario was that client who had set their site's Temp-folder to the site's administrator. When Joomla run a temporary files clean up it did exactly as it told, removing all files and folders from the temp-folder, i.e. wiped out the entire backend.

On the other hand, manually excluded folders (yellow icons) are just that: a human operator with access to Akeeba Backup did that. The easy part of the solution is to remove the exclusion manually. The hard of the solution is finding out who, when, and why did they do that. It's doubly hard on sites where you're the only administrator because you know who did it, but you'll never know when or why. It happens to everybody.

Regarding Admin Tools, it does not require you to add the restored domain to the Allowed Sites before the backup unless you have a really old version of Akeeba Backup. Support for handling Admin Tools' Allowed Domains feature during restoration was added in January 2022, and the code was revised for newer Admin Tools version in April 2024. Kindly remember you never told me which version you have.