I havent been there lately, traditionally its plowed up to the fish hatchery. As commented by many, the gate is generally left open to enable them to plow it. If it is closed, it is rarely if ever locked and worse case is that you would need to go knock on the door of the houses near the gate.
There are no guarantees that the road is plowed to the Bunnel Notch trail head from the actual hatchery buildings, but it normally is. (it was last winter)