Performing everything as rehearsed is not jazz. When you listen to jazz, you are hearing raw ideas being given substance even as they must adapt to the musical setting. Many musicians describe a sense of ideas flowing through them rather than from them. In this sense, jazz is sacred as is all of God's work. Jazz imitates God's creation - ever evolving.
Jazz involves cooperation - It is a community that performs and receives jazz.
Each musician is expected to both support the others and shine as an individual. Listeners inspire through their reactions to complete the circle. Church is about community. Christians are a community. It's one reason I can feel at home in your church though I'm a guest.