Cool, I didn't realise that Greek was a phonetic alphabet!

I think there is a difference between an alphabet that one uses from birth, and one that you try to teach to adults. In the former case, you're starting from scratch. In the latter case, you've got your old alphabet confusing things.

Evidently, phonetic alphabets are possible; but for an adult to learn to use one it would take practice and effort, and many people would probably be fairly resistant to that ("our old one is perfectly good", "if it ain't broke, don't fix it" etc.).

That's true, but everyone is resistant to everything. A problem with English is that there are more homophones than there are in Greek. When you see a Greek word, you can pronounce it and, in the vast majority of times, you know what it means. Not so in English, where there are many more homophones. Context helps, though.