This isn't just because I don't much like imports from American English into British English, but is because "Hi" is so much less efficient as a greeting than alternatives like "Hello" or "Good morning" - especially if you're making a phone call and can't see the person who's answered it.
Some of the early work in conversation analysis took a detailed look at greeting sequences, and came up with the idea that the first thing we do when we hear a voice on the other end of a phone is a 'voice recognition test'.
The rule is: if you can recognise the voice, you should immediately let the other person know that you've recognised who it is.
So, if someone answers the phone by saying "Neasden 456789", you have quite an extended voice sample (9 syllables) on which to do the voice recognition test before the answerer reaches the end of the number. By then, if you have recognised it, you should promptly acknowledge the fact by saying something along the lines of "Hello Ron" or "Hello Mr Knee."
The advantage of this for Mr Knee is that he doesn't have to go to the trouble of introducing himself or explaining who he is or where he's from, because you've already established that you know perfectly well who he is.
Like quite a lot of rules in conversation, the rule has an 'if you can' clause to it. In other words, there's a preference for showing instant recognition over failing to show recognition - so the first option is to show that you've recognised the answerer - if you can.
This is why the word "Hi" is such an inefficient or inadequate form of greeting when you can't see the person who's speaking - for the obvious reason that a single syllable on its own may not be enough for you to be sure who it is within the split second before they've finished. As a result, you'll have to admit to them that you didn't recognise their voice, which can sometimes have quite embarrassing consequences.
This might seem a rather trivial reason for suggesting that multi-syllable words and phrases like "Hello" and "Good morning" are more efficient than "Hi". But it's not at all trivial when you're on the phone, or if you happen to be blind or visually impaired.
I know this because the person I've heard objecting most strongly about people greeting him with "Hi" is someone who's been blind from birth. What's more, the reason he gives for detesting it so much is precisely because it doesn't give him enough time to know who it is that's speaking to him - and makes him feel impolite for having to confess that he'd failed to recognise them.