To answer your direct question, no – The existence of a “cause of the cause” doesn’t disqualify the secondary cause. In your example, based on my experience, *either* the peat burning or the ozone depletion might be given as a correct answer in a weaken question, but you wouldn’t see both given on the same question with the ozone answer being considered wrong.
As for answer choice (C) on this question, it’s a typical example of something seen often on strengthen or weaken questions with a cause and effect argument. If the cause and effect are both present (or both absent), then the argument is strengthened. I think you might have just over-thought this one. For more on strengthen/weaken questions and the cause/effect argument, see: http://thelsattrainer.academy/forums/topic/strengthenweaken-questions/
That post also covers two other common ways in which strengthen/weaken questions tend to resolve on the LSAT.
Hope this helps,