Oops, I thought your claim was about the consonant sound /b/ vs /v/. I had the British /bi:tə/ in my mind, and forgot that Americans used /beɪtə/, which I agree is closer to the American pronunciation if your 'ay's are not diphthongised.
Funny enough, I went to double-check the IPA and realized the textbook classical Attic should be reconstructed as /ɛ/, so /bɛːta/ anyway. Which is still closer to the American version as both are open front vowels.
It turns out that while /bɛːta/ is the old academic reconstruction, statistical analyses of spelling mistakes from then shows that Athenians had already closed that vowel to /e:/ or even all the way to the modern /i:/ sound as early as 500 BC. So the how they spoke daily was even messier.
You're right, I guess American English is too common and I didn't consider that British English pronounce it differently (and one step closer to modern Greek)
Attic/Athenian Greek were considered a bit weird by other Greeks at the time, especially with replacing "ss" with "tt". But if there was nothing else to connect us modern Greeks to ancient Greeks, the constant infighting and bickering would be enough :)
Doric Greek had replaced η with α in the same time period (eg ή ταν ή επί τας which would be ή την ή επί της in other ancient Greek dialects of the time.
