That is searching the post meta fields but it is not adding the term name to a meta field. There is nothing in that code that indicates to me that the term name has been added to post meta.
You would need to create an acf/save_post action. In your function you would need to get the terms for the post or get the value from a taxonomy field if that is what you are using. Then you would need to use update_post_meta() to update/add the term name(s) to the post.
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