The parameter cards_fields=checkItemStates in GET /1/search returns only those check items, that are completed. I think it looks like a bug, rather than a feature.
It looks like checkItemStates is always returned as a field. And you’re correct, it only ever returns the completed items. As I mentioned in the other thread, we pass the request to an ElasticSearch instance and then return the response from ES.
I believe our recommendation here is that if you need the checklists on a card that has been returned, you should make a GET request directly to the card’s resource: https://developer.atlassian.com/cloud/trello/rest/api-group-cards/#api-cards-id-checklists-get