In the above, $x, y\in V$ (not necessarily for all points in the vertex set) and $\tau_y$ is the first hitting time of $y$ for a simple random walk on the vertex set $V$.
My gut feeling is that it is not true because we may not have symmetry in the graph but I cannot think of a simple example.
I thought maybe adding the point $ x = (1/2, 1/2, 1/2)$ to the graph of $\mathbb{Z}^3$ by an edge from $y = (0,0, 0)$ and then due to the transience of $\mathbb{Z}^3$ we have the transience of the new graph even though $\mathbb{P}_{x}(\tau_y < \infty) = 1$.