If in your sentence "If the clause is within a sentence, both dashes are necessary." the comma is optional, as you indicate, then it is a poor example of what effect the insertion of a clause should have on that comma.
If such an extra-information clause appears within of a sentence at a position where no comma (or semicolon or whatever) need otherwise appear, then of course you need a dash at each end of the clause, but such a clause can also appear at the end, in which case the full stop swallows the closing dash. Similarly, one might want to insert a clause before an existing mandatory comma or semicolon, and surely then the better course of action would be to let the dash be swallowed in the same way, rather than let it do the swallowing.
I can't see how in the following enumeration:
On our afternoon walk we went past the watermill, the stables - about which I must tell you a funny story, the duck pond, and the tractor shed.
you could possibly replace the comma after 'story' by a dash. Still, this is probably a good example of where the use of a dash is a bad idea, and parentheses would be better:
On our afternoon walk we went past the watermill, the stables (about which I must tell you a funny story), the duck pond, and the tractor shed.