I did want to address this, sorry for leaving it out a second time. I don’t think that Cage ever said nothing, it’s that he said exactly what was there in the work itself.
Thinking nothing as a concept is still very much “thinking”. It’s still very different to “not thinking” or having “nothing to think”.
When one consigns one’s thoughts to the Cloud of Forgetting, in order to open oneself to the Cloud of Unknowing, it is not to have this cloud or the concept of nothing in one’s thoughts, it is literally to consign those thoughts. 
[on the other hand… not a perfect analogy; engagement in creative work is moving out into the between, not a total abdication, otherwise there would not be dinergy. it is still useful to maintain the distinction between action and contemplation. still, the same basic performatives apply… which hints at the grounding of the active life in the contemplative.]
The thing with Cage is it’s never anything specifically modern he’s doing, he is simply tapping into various currents of ancient wisdom, whether from Zen, Christian mysticism, Gurdjieff or other sources. They live on again in new ways in his work.
(I’m also sorry if this is cryptic… it’s the best I can do.)