Transforming an arbitrary cover into a vertex cover
If no points in $C$ lie exactly on the mid-point of an edge in $\mathcal{G}$, then it suffices to associate each point in $C$ to the nearest vertex in $\mathcal{G}$. I will leave it as an exercise to the reader to prove this (hint: prove by contradiction).

On the other hand, if points in $C$ are allowed to lie on the mid-point of edges, then we can provide a counter-example:



The blue lines and circles are $\mathcal{G}$ and the red crosses are $C$.

EDITED TO ADD: An example with a biconnected graph