Why is the social value curve NOT the supply curve shifted downwards? Abbreviate 'Social Value' as SV. In some other resources such as this, the SV curve is known as the Social Marginal Benefit curve. 

The following quotes explain [1.] why the SV curve is the demand curve shifted upwards, but they don't explain [2.] why the SV curve is NOT the supply curve shifted downwards. 

Here's my rationale for [2.]: Assume $SV &gt; 0$. Because SV benefits society, the suppliers of this SV can consider the SV supplied, as follows:
Suppliers' true cost  $=$ their original costs $-$ SV, $\qquad$ for all quantities supplied.
So the new supply curve must underlie, and be parallel to,   the old supply curve.   

Source: pp 179-180, Modern Principles of Economics (2 ed, 2011) by Tyler Cowen, Alex Tabarrok


  [See Figure 10.3.] 
  The social value curve counts all the benefits of vaccine
  use, the private value plus the external benefits, so the efficient quantity is
  found where the social value curve intersects the supply curve.
  
  A final way of illustrating the underuse of vaccines is to notice that if people
  who got a flu shot did receive all the benefits of vaccination, then their demand curve would shift upward by $20 and would be the same as the social
  value curve. ...
  
  The last way of thinking about the problem of external benefits also suggests
  one potential solution. If every time someone was vaccinated, they were given
  a subsidy of $\$20$, the monetary equivalent of the external benefit, they would
  demand the amount $Q_{Efficient}$. Recall from Chapter 6 that we can analyze a subsidy
  by shifting up the demand curve by the amount of the subsidy. Thus, in
  Figure 10.3, notice that a subsidy set equal to the level of the external benefit
  would shift the demand curve up and increase the market quantity from $Q_{Market}$
  to $Q_{Efficient}$.


Footnote: I was reading p 201, Principles of Microeconomics (7 Ed, 2014) by N G Mankiw, when I encountered this problem. I don't quote Mankiw's book because it's brusquer than Cowen's above.
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.