The very wording "I would like you to [do something]..." conveys the idea of "that", so not we do not normally use "that" after WANT/WOULD LIKE.
That boils down to one thing they call convention, or at least that is my guess.
One more thing: not using "that" turns the sentence more economical, plus in a similar way to the topic of object pronouns (in English, I mean), the omission of THAT here doesn´t makes the sentence impossible to understand. Only that in this case (with I would like...) it´s not optional, you just don´t use the relative pronoun.
I used the analogy just to make you see that the rule of not using it doesn´t ruin the comprehension; but fact is, one doesn´t use it. I peg it to convention (along with pragmatism - since they use simpler structures, in an economical way).
Anyway, let´s leave the last word for the grammarians and more advanced learners.
Topics used in my analogy/explanation.https://languageonschools.com/free-english-lessons/the-conditional/hope-expect-want-would-like/http://englishharmony.com/omitting-relative-pronouns/