Then what should they teach, if they can’t teach those things?

Vocal technique! Just vocal technique! Teachers shouldn’t substitute the peripheral aspects of style interpretation for basic vocal technique. It’s a totally different thing. Most pop singing has one thing in common: it’s on a conversational level. Opera and other forms of traditional styles are not always that way, but you must still be able to go into your head voice without leaving your speech level. Most students and teachers who sing opera base their modern idea of operatic tone on a concept of a “woofy,” overproduced sound, which is dangerous to the health and longevity of the voice. What is interesting is that the best opera singers (of yesterday and today) sing in a clear, speech-level manner that lets you understand their words all the way through their ranges. This is the same ideal that people listen for in any type of good singing.