Plugin not found: "The "search_api_autocomplete_search" entity type does not exist."

Sorry, that is much too unspecific to be able to tell you more. It just seems more like a problem in Core, or maybe with some other module – the search_api_autocomplete_search entity type is defined in this module, so we should always be free to use it and safe to assume it exists. Maybe some code somewhere is trying to access it at a time during installation/importing when it shouldn’t, but impossible to tell when and where. Maybe try debugging the stack trace of that fatal error, does that tell you anything?

Confirm this message appears when configuration import `drush cim` on a Drupal site where the module is not in composer.json
When it gets a composer require as stated on the module page, all works okay for me.