Why do the Japanese say Bye bye?

I know some gaijin (foreigners) who find this really annoying, an irritation that gets even worse when they meet a Japanese person who doesn’t even know that it comes from English. “Why can’t they just use a Japanese word?” they ask, of course already knowing the word sayonara. Well, sayonara for one won’t cut it, as it is more like “Farewell” or “Take care” than it is to a simple “Bye bye”.

 There’s “domo”, but it also means “Hi” and “Thanks”, and anyway doesn’t seem to match every situation. Ditto for ja ne. The one situation they especially don’t match is babytalk, being that waving is one of the first things a baby picks up but none of those expressions quite work to prompt that waving from them. Perhaps because of this, almost every country I have lived in have also used “Bye bye”.

Alternatively, I wonder if in some countries it could have come through telephoning, in the way that Hello came in French and some other languages.

Another question occurs to me writing that- why does Domo have some many meanings?