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You should not be removing files manually. Joomla extensions are more than files. There are also database records.

What you've done is actually breaking Joomla. As far as Joomla knows, the extension is already installed, therefore it tries to to an update. However, its database tables do not exist which is why you are getting a failure when Joomla tries to update the non-existent database tables. This will, of course, never work.

For each extension you removed –package, component, and every plugin– you have to remove the corresponding #__extensions record, noting down the extension ID. Then, you have to also remove the corresponding #__schema record, if one exists (there's definitely one for the component). This will only break your site a little bit instead of completely (it makes a mess of the #__assets table, but at this point there's not much else you can do given what you have done).

If you want to do things the right way, you should instead restore the latest backup and uninstall the package extension from Joomla's extensions management page. Then, you can install Akeeba Backup afresh.

This is not specific to Akeeba Backup. It applies to all extensions delivered as an extension of the Package type which is the correct way to deliver compound extensions (extensions which have more than one of a component, plugin, module, template, files, language extension or a combination thereof) since Joomla 1.6 was released 16 years ago.

Yeah, I know you know, but since it's a public ticket I have to explain why it was butchered for the people reading this who don't know what we know. You know? :D

The best you could do is try to remove the #__schemas and #__extensions records for the component. I imagine that this would be the least invasive way to allow the reinstallation of the extension.