My friends are still worrying over classical singers doing jazz and pop. I have only thought that Eileen Farrell was even marginally successful at this.
The way it works is this. Vocal technique also has different styles. A classical singer learns to hold his larynx in a position that allows the voice to maximize its vibrato. This is what is wanted in the opera world. What is desired is the maximum richness and the maximum flexibility at the same time. Except for Wagnerians who can do without flexibility.
Up until the recent ornamentation phase flexibility was not a requirement for a pop singer of any genre. A pop singer needs completely successful mixing in the center which is usually achieved by singing low, flattening out the tone, and not pushing the larynx down. This is a completely different technique.
I know that as a young person I had a voice teacher who decided I was a true contralto. She spent many hours teaching me to push my larynx down to get a contralto-like sound. I couldn't sing pop any more. Mixing in the center got a lot harder. I have since decided that she was wrong and completely screwed me up.
Then there is the style issue. The right style for Wagner is not the right style for Gershwin. It's also not the right style for Puccini.
You can mix these genres only if you are happy with being bad in all but one and not truly great at any. I haven't heard Renée Fleming's jazz. I'll look into it. I liked her operatic "Over the rainbow" because she didn't try to pretend she isn't an opera singer.
The wisest singers in any genre will focus on developing a style and technique that completely suits them and then limit this to a repertoire where it is suitable. Ella Fitzgerald didn't branch out into opera. Perfection in music is usually combined with unity of purpose.