As a native speaker fussy about language, leaving out unnecessary "that"s is something I wish far more native speakers would do. I disagree the sentence sounds wrong, of course, and think including a "that" sounds far more wrong, and should only be done to disambiguate, and only if you cannot reword the sentence to resolve the ambiguity without a "that".

"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.