I think reading is somewhat important to becoming a writer, but it isn't top priority. Reading essentially introduces you to new ideas in such a format that you can take time to digest and go over, and also influences one's style. It is essential for all writers, but how much reading is required depends on each writer, I would assume.
More important, I think, is what we find in living life itself. Once again, this is more for certain writers than others, but I think living one's own experiences is more important for 'good' writing and than reading of another's for any writer.
Reading too much can make one absent-minded and not unique. Life is the only thing where any artist can be original. Artists are essentially thieves always out for stealing. Originality exists only as a synthesis of ideas, and while the ideas may come from reading that ability to synthesize, to make original art, comes from life: the beauty of life.
I say for fiction writers and historical non-fiction writers, reading is very important. But for one who writes memoirs, then reading other memoirs really isn't that important aside from developing style.