What is it that makes bye-bye so inherently juvenile compared to goodbye, or simply bye?
I have to imagine it’s the repetition, isn’t it? Surely it reminds us of babies, repeating simple sounds. Think of how we stereotype baby speech: goo goo, ga ga. Bye-bye.
It’s also probably part of the general trend in English of avoiding statements that are too abrupt. The monosyllabic bye can seem a bit rude. Think about how we usually actually say it: byyeee! We always elongate it to sound less rude (unless we want to show someone we’re annoyed).
Imagine telling a young child to say bye to someone, rather than bye-bye. Weird, eh?
English-Language Thoughts 
South Koreans often say bye-bye in English. The first time I lived in South Korea, I consulted a musculo-skeletal doctor and a physio working for him. At the end of every consultation, the doctor said bye-bye.
By coincidence, earlier today I was talking on the phone with someone from a training college. At the end of the conversation, I said byyeeee, and the other person said bye-bye.
Perhaps people don’t say goodbye (as much) now. They certainly don’t say God be w’ye or God be with ye. Saying God be with ye would mark you as remarkably odd.
A student once asked about hello, specifically that it contains ‘hell’: https://neverpureandrarelysimple.wordpress.com/2015/06/06/hello/
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I imagine ‘goodbye’ is almost completely dead in conversational English. I’m trying to imagine using it, and it would sound so formal as to be deliberately cold.
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No one does juvenile like Our Glorious Leader.
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