In reply to:

Try pronouncing the vowel sounds from bet and bit together very quickly without any intervening h or glottal stop and see what you get.

I'm afraid this coarse colonial is beaten! For me "bit" has only a schwa, and running it together with "bet" doesn't come close to the way I pronounce rain. I have had a lot of practice at diphthongs, Maori is full of them, actually full of separate vowels which must be said together at speed. Despite this, I failed the exercise you suggested miserably.