From a (quite) theoretical point of view, if all you are doing is buying puts and calls across all strikes $0\leq K \leq u$ up to some maximum strike $u$, you are aggregating the corresponding call deltas $N(d_1(K))$ and put deltas $N(d_1(K))-1$ across strikes. Per strike position, this results in a total delta of $2N(d_1(K))-1$.
Now, let us assume a dense ...
A bit too long for a comment.
TLDR: It's hard because there is interaction between your strategy and market movements.
If you're executing a option strategy aiming for certain exposures to the Greeks I don't think a statistical approach makes sense. You should be getting the exposure you're aiming for.
If you're providing liquidity a statistical approach ...
Correct. You're starting with specifying the scenario where all rates move parallel. The correlation implicit there is 100%. There is nothing stochastic left. The gamma to a parallel shift is as cited originally. No matrix is needed.