it is possible to teach people to sing without teaching them style, you teach them how to sing in tune how to have good diction, you can show them how to gradually improve their size of vocal range, so yes it is possible to teach technique without imposing personal views. it is possible to teach the m breathing exrcises and how to use the diaphragm to get better control of air.
none of this has anything to do with style,
you can also teach them how to interpret a song to bring out the story, in my opinion the best way to do this is by taking a song and showing 3 different interpretations, that way a student can be conscious of different styles, this is in my opinion is a preferable alternative to playing a recording of one singer and saying this is the best way to do it