Looking at your original code, see my comments in the code
$field_key = "average_review_score";
$value = "8";
$post_id = "3";
$get_av = get_field( "average_review_score", $post_id );
echo $get_av;
// this is updating the post with an ID of 3
// it is not updating a term in a taxonomy
update_field( $field_key, $value, $post_id );
If your intent is to update a field that is attached to a term in a taxonomy then you need to use a $post_id that represents that term. See “Getting values from other places” on this page https://www.advancedcustomfields.com/resources/get_field/
