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Phrasal verbs are listed under the main verb. I never ever had a problem with that. As a native speaker sometimes I still have to search for some in some strange context.

These are called idiomatic phrases, and many (all natural?) languages have them, and, yes, they are pitfalls for language learners.

These particular examples are figures of speech, so "shut" in "shut up" still means the same thing it would mean in "shut the door." And "up" is used the same way as "cover up."

So the issue is just that this is figurative language, and you have to know that a kickoff is the beginning of certain sports, for example. It's more of a cultural issue than something a dictionary needs to fix.