Snowboards definitely have steering, you twist the board and shift your weight to manipulate how the edges contact the snow, it's just not quiet as explicit as a bikes front wheel. But whether it's a bike, a board, or literally any moving thing on land, the steering happens because you applied a lateral force to the ground and an equal and opposite force was applied back to you.

The snowboard uses different methods of applying that force, but other than that it's the same concept as described in my first comment: Greater speed allows more subtle corrections to take effect more quickly.

Now the snowboard does have a wider contact area with the ground, but that really only helps you on flat ground at very low speed, or standstill. Advanced boarders will carve transitioning from edge to edge most of the time.