Cazzanut is not opera although it is uses similar techniques.
Cazzanut originated in the synagogues starting off as nusach hatfila with chazunam often adding their own peices and original compositions,as chazzanut developed it did borrow some influences from opera but as any musicologist will tell you Chazzanut is a genre on itd own.
As for Yossele Rosenblatt, the New York Opera House recognised his potential as an opera singer but he refused their offer despite being assured that he would not have to perform on shabbos or with female singers.
Most pop music styles trace their roots to Jazz which traces it’s roots to the pagan tribes in Africa. So there is quite a strong argument to say that while some influences are okay others are not and can never be considered Jewish music.