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.
You're confusing different terminology. There is a traditional pronunciation. And there's a classical pronunciation. But the traditional pronunciation is not classical.
For Latin, there's a handy wikipedia page telling you all about the difference: https://en.wikipedia.org/wiki/Traditional_English_pronunciat... .