I question whether it is absolutely necessary that the audience know how a piece is composed. You don't have to deeply comprehend something in order to enjoy it. For example, you can admire a great painting and yet know very little about the technique of painting. There may always be some works that the audience never understands, just as some late Picasso works may never be fully appreciated. But the pieces shouldn't be banned from the stage because of this, just as Picasso's paintings shouldn't be taken from the museum wall.
Some pieces are more difficult to convey to the audience than others.
Some require more study and familiarity, but this does not necessarily mean that there is something wrong with the piece. Just as some books require careful and deliberate reading, certain music requires repeated hearings and study as well. We do our best to communicate the piece to the audience. If they get it, we have accomplished something truly wonderful. I would say this is true of more than just contemporary music.