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.