Crown Rule Reduction In Parameterized Complexity - Vertex Cover - Notion Question
I agree with the OP: $\mathcal{G} - \mathcal{C} = \mathcal{G}'$.

I assume the author of the paper wanted to define $\mathcal{G}'$ as $\mathcal{G} - S_{\mathcal{C}}$. With this modification, the next sentence in the paper makes sense:


  One can easily see that $\#vc(\mathcal{G}',k-|S_{\mathcal{C}}|)$ equals the number of size $k$ vertex covers $S$ in $\mathcal{G}$ with $S \supseteq S_{\mathcal{C}}$.


Here, $\#vc(\mathcal{G},k)$ denotes the number of size $k$ vertex covers in $\mathcal{G}$.