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")
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.
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.
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.