No facet source with core search

Maybe update the error message with something like:

Before adding a facet, there should be a facet source. Facet sources can be:
- Drupal core's search.
- A view based on a Search API index with a page display.
- A page from the search_api_page module.

Having the same issue on Drupal 9.5.8 with Facets 2.0.6. Core search enabled, indexed, and correctly outputting to /search/node. Yet Facets config reports no facet sources, and admin/config/search/facets/add-facet results in the same. No errors or warnings in Watchdog.

Same here, do we know if the module actually supports core search

It no longer does out of the box, you can find the old code in the facets_core_search module, but this is not actively maintained. We however suggest you use Search API instead.

Both the project page and the README say that Facets work with Drupal Core Search, so they probably should be updated.

Facets 3

Facets 3.x only supports Search API, and introduces "Facets exposed filters" next to "Facet blocks".
We recommend using Facets exposed filters for new projects, when using Facets + Views
Using "Facets exposed filters" in Facets 3 has full AJAX support.

Facets 3 will not support AJAX for "Facet blocks".

Facets 2

Facets 2 works with Drupal Core Search and Search API.
It is kept available for existing projects using Facets 2 + Ajax as there is no upgrade path for this setup. Updating from 2 to 3 without AJAX support is possible.

I think this is very clear, for facets 2 you need the additional core search module as mentioned here, if this doesn't work I think we should file a bug report there. Readme update in #3506177