What you call "classical pronunciation" is really at best an approximation of the ancient Greek pronunciation, but mixed heavily with English (after some frolicking around in Latin). As far as I know, this is limited to English speakers only.
For example, π is pronounced "πι", or probably closed to "pee" in modern and in ancient Greek. It's never pronounced like "pie". Same with all letters that end with "i", for example "φ,χ,ψ" (pronounced as phee, chee, psee, never rhyming with pie). T (τ) was never pronounced as "ta-oo", either, not in ancient nor modern Greek.
There are differences between modern and ancient Greek of course. For example "β" (beta), originally pronounced more like it's now in English, only with a longer "e", while in modern Greek it's more like "vita")
I had a native Italian professor who said "β" more like "vita". One day he tried to write "ξ" on the whiteboard and said "whatever the hell this is?". I was the only person who spoke up.
> For example, π is pronounced "πι", or probably close[?] to "pee" in modern and in ancient Greek.
No. In ancient Greek, π contrasts with φ. Φ is the one that indicates the sound an English speaker would hear as "p"; it's the one you would pronounce "pee". You'd hear the name of π as "bee".
> T (τ) was never pronounced as "ta-oo", either, not in ancient nor modern Greek.
That's exactly how it was pronounced in ancient Greek (modulo the same issue as π), unless you meant to indicate a disyllabic pronunciation.
I'm well aware it is an approximation, but there is a traditional classical pronunciation in use as there is with Latin (or Sanskrit, Pali and classical Hebrew), which is still in use whether or not it is authentic.
The modern academic consensus is that "η" was likely pronounced like the "e" in "met" but longer. In IPA, it'd be /e:/. And thus "β" as /be:ta/. What you are saying is how it is done in modern Greek though.