I think what you’re looking for is an explanation of the “threshold of motion” which is the point where the tires slip and go from static to kinetic friction. On certain snow and ice conditions the available static friction is low and kinetic even lower. So your car is trying not to apply very much torque to the tires. And starting in second gear is easier for the vehicle to very gently apply the power to the tires in automatic transmissions because of how they function internally.
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