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There needn't be a contradiction; the history of London waterways is usually pretty complicated as they were constantly worked on, in general canals were a lucrative enterprise only for a fairly short time because of railways, and because while many companies went bust, others took over (often with the same workers, of course). When canals were filled in, this often happened in stages, etc.
The Wikipedia page shows the complexity of the history of the GSC:
OK so maybe that was some other canal I was being told about... perhaps the one that ran though Brockley was some other spur?