"I wish you'd do the dishes" sounds fine to most native English speakers, except those who are extremely fussy about language. Leaving out "that" in this sentence will sound wrong to most native speakers (though we can easily gloss over it when reading).
"I wish you'd do the dishes" is fine. Far better than "I wish that you'd do the dishes". The "that" is useless. It's contextually clear what you're wishing, with no misunderstanding.
Many "that"s hide poor sentence structure and are a sign of lazy writing/thinking. Take the following:
- I realize now I need a car.
There is some ambiguity about what exactly I mean. I could use "that" to clear things up:
- I realize that now I need a car.
- I realize now that I need a car.
But that helps me spot the problem isn't the lack of "that", it's the "now", and I subtly alter the sentence's meaning based on where I place it:
- I realize I now need a car.
- I realize I need a car now.
- I now realize I need a car.
I should pick the sentence that most clearly communicates my meaning.
It should also be noted that leaving out "that" (if grammatical correct) can sometime sound wrong to native speakers, but that varies from accent and region since there is no standard (at least as far as I know).
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