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I meant they can recognise it, not simply as being music, but as something they have heard and enjoyed before. Whereas, if you get somebody to listen to a bit of Kagel, chances are they won't have heard it before, and won't enjoy it.

Hall of Fame Top 300 - Numbers 1 to 80 - Classic FM

This is a list as voted for by the British public (the orchestral music loving side public). Almost the same pieces get voted into the top 100 every year, because they are popular pieces of music and remain so today. There are a few composers on the list I don't recognise, but I highly doubt they are atonal composers.
Of course, I personally disagree with many of the choices on this list, but essentially this list was voted for by the people who go to orchestral music concerts. That there doesn't appear to be a single atonal piece on the list implies that generally, the public do not want to listen to atonal works. Whether you think this is a good thing or not is barely relevent.
I recognise that in England, atonality has always been frowned upon, although there have been many English atonal composers, who simply havn't had much or any public success.

Debussy could have said this, Williams could have said this, Prokofiev could have said this! I could say it right now, but I won't because it isn't true!