The first seeds of doubt as to the authenticity of this 'quote' surely arise from the question of whether GBS would have used the word "closet" rather than "cupboard".
According to Good Reads, Shaw wrote in Immaturity: "If you cannot get rid of the family skeleton, you may as well make it dance." This was his first completed novel, written in 1879, which (according to Wikipedia) "was too grim to appeal to publishers and did not appear until the 1930s."
So it wasn't something he said, but it seems he did at least write something along the same lines. Presumably it was said by one of his characters, so it's not actually GBS speaking as himself.
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