How can I allow users to edit text that will be displayed on a custom post type archive page?

What your asking for is basically what is called a sticky post for archives (or a taxonomy), a widget or a content block.

Either way you will have to write some code. To tie it into the admin bar is pretty easy, for example using admin_bar_menu, it has several hooks.

Personally I would just create another CPT called “sticky archive posts” (or something) and use that with the WP Query to create a sticky archive post and link it in the admin bar.