Why is the social value curve NOT the supply curve shifted downwards?
Here is how I interpret your question: "Why when modeling positive externalities do we shift the marginal benefit curve upward instead of shifting the marginal cost curve downward?"

Either way of thinking about it is fine. Positive externalities can be thought of as resulting in a social marginal benefit curve that's above the private marginal benefit curve, or as resulting in a social marginal cost curve that's below the private marginal cost curve. Which side you place it on is only a matter of accounting, since 

Private Marginal Benefit + Externality - Private Marginal Cost = Private Marginal Benefit - (Private Marginal Cost - Externality).

Your question doesn't make sense in some ways though and I think it's important you understand why. You say "Because SV benefits society, the suppliers of this SV can consider the SV supplied, as follows: Suppliers' true cost = their original costs − SV, for all quantities supplied.
So the new supply curve must underlie, and be parallel to, the old supply curve."

The positive externality in this case cannot shift the producer's supply curve downward, because the producer's supply curve is his private marginal cost. Importantly, externalities only affect social marginal cost. 

Remember: The supply curve is a function that tells you what quantity a supplier will produce at any given price, whereas the marginal cost curve is the cost per unit of producing an additional unit. These two things only happen to coincide in the absence of externalities.