Well it doesn't manage the update sites, so the text is misleading, you are right.

But it is used to manage the update site ;-)

We use this extension to link the update site for the core to an extension id.

And it has helper routines for managing update sites:

https://github.com/joomla/joomla-cms/blob/4.0-dev/plugins/extension/joomla/joomla.php#L64

https://github.com/joomla/joomla-cms/blob/4.0-dev/plugins/extension/joomla/joomla.php#L147

...

So maybe the description is not that wrong.

Maybe: "Helps managing the update sites of extensions"?

My problem is that the original and your proposed description dont match the title

Extension - Finder
Manage the common words in Smart Search.

Extension - Namespace Updater
Automatically builds and updates the administrator/cache/autoload_psr4.php file that is used to autoload extensions.

Extension - Joomla
Manage the update sites for extensions.

Hopefully you can see by my highlights that the title of the first two extension plugins refers to their specific tasks. So I would assume from the title that Extension - Joomla is managing the update sites for the core of joomla and not for extensions.

@richard67 is saying that it is managing the update sites for core AND extensions. If thats correct then I would suggest
Extension - Updates
Manage the update sites. (replace the word manage with something more specific perhaps)

I

How about
Manage the update sites for extensions and Joomla.