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Yeah, I can tell you what happened. Remember, the migration from Joomla! 3 to 4 happened for the majority of our clients back in September to November 2021. I've seen it all, two years ago. This new wave of upgrades is just deja vu for me :)

The latest version for Joomla! 3 is Akeeba Backup 8 which is meant to act as a “bridge” version upon upgrade to Joomla! 4. Meaning, it works perfectly file on Joomla! 3, but on Joomla! 4 it will not let you do anything with it by default. It shows you a page which reads the following (the Joomla version number may be different on your installation):

You are currently using Akeeba Backup 8. This version is only meant to allow you to upgrade from Joomla 3 to Joomla 4 without losing your backup archives and settings. It is not supported for taking on restoring backups with Joomla 4.4.2.

You need to download and install Akeeba Backup 9, our Joomla 4 native version of Akeeba Backup. It is fully supported for use on Joomla 4.

After installing Akeeba Backup 9 please click on Components, Akeeba Backup for Joomla!, Control Panel from Joomla's sidebar and follow the instructions on your screen to migrate your settings and your backups from Akeeba Backup 8.

Unfortunately, you seem to have installed Akeeba Backup 9 and uninstalled Akeeba Backup 8 without going through the migration first. Hence the loss of settings and backups.  When you uninstall Akeeba Backup Joomla! deletes its folders and database tables, therefore it deletes the backup history (#__ak_stats), the backup settings (#__ak_storage), and the backup archive files.

If you follow the instructions in the message, upon installing Akeeba Backup 9 you go to Components, Akeeba Backup for Joomla!. A big box at the top of the page tells you that you need to migrate the settings and backups from Akeeba Backup 8 and gives you a button to do that. You click that button and the migration takes place automatically, usually in a matter of a couple of seconds. Then, it gives you instructions of uninstalling Akeeba Backup 8.

Noting the comment made en passant about restoring your site to Joomla! 3, I have to note that things are not as you think. Had you needed to restore your Joomla! 3 site, you would not have been able to use the backend interface. As documented, you'd need to delete all files and folders, upload the backup archive file(s) and Kickstart at the site's root, then run Kickstart to extract it, and tell ANGIE (the restoration script) to delete all tables during the database restoration. Anything else would result in a mix of Joomla! 3 and 4 which makes for a broken site. This has always been true across major Joomla! releases (2.x to 3.x, 3.x to 4.x, 4.x to 5.x, or vice versa), and more often than not minor releases as well (e.g. 3.6 to 3.7 and vice versa). I am commenting that even though it's not part of what you asked in case you have another site you are migrating, to save you some frustration.