The "that" most certainly should not be in the sentence about dishes—and by most certainly, I most certainly mean subjectively most certainly because it's just a damn awful sentence. You've given a great example of why skipping an unnecessary "that" is as important as not omitting "that" where necessary.

While the article's mistake is a result of omitting a necessary "that", this example about dishes is a case of inserting one that is ugly, unnecessary, and actually ruins the sentence flow.

My other comment to a sibling explains more: https://news.ycombinator.com/item?id=11714645