From my reading, the point is that the glass of water doesn't really "have a temperature" any more, since the cooling transition means it's no longer in thermal equilibrium.
That is, you could perhaps model the bulk of water as having a temperature field, and clearly every point in that field passes through the starting temperature of the initially-cool system, but the gradient landscape is vastly different.
The point is that the state of the system consists of more than its average temperature. It's not just "catching up" to the state of the initially-cold water, it's going through other configurations.
That is, you could perhaps model the bulk of water as having a temperature field, and clearly every point in that field passes through the starting temperature of the initially-cool system, but the gradient landscape is vastly different.