I'm talking about the vowel sound, not the consonant sound. Of course φ was more like an aspirated π, that's why the letter φ is transliterated as ph (p with an aspiration mark which is the h). χ was also originally transliterated as kh, as it sounded more than an aspirated "k". I'm not sure about the consonant sound of "ψ" though. Certainly not "s" of course (the "p" in "ps" is not silent in modern Greek, and it most certainly wasn't silent in ancient Greek either).

π, φ, χ, ψ never rhymed with pie though. That was my focus there.

Regarding the "αυ" sound: Same as all the original diphthongs such as "ου", "αι", "οι" etc. sounds, the "ι", "υ" etc were mostly supplemental/modifiers to the first vowel (also υ sounded more like e.g. modern German ü or indeed modern German y Ι guess). They were indeed never disyllabic.

EDIT: I think the transformation of some diphthongs had already started by the time the Roman empire conquered Greece, so ta-oo might have been closer to the pronunciation at that time. But Roman times are not classical times, they are after the Hellenistic times which changed so much already (I think iotacisation happened during that time?).

Koine Greek started off as more like Ancient Greek pronunciation and ended up as Modern Greek pronunciation.