I like the comparison to horses. Steam engines and trains threatened to replace horses in the mid 1800s. A lot of things that were previously done by horses were replaced with trains and steam engines.
Yet the horse population grew, and cities remained full of horses. There were countless transportation innovations in the 1800s, from canals to streetcars and omnibuses to bicycles. But the horse population kept growing and showed no signs of being threatened by this "automation". Whenever something took a horses job, there will always be other jobs that technology can't do yet, right?
Then the car was invented, and within 2 decades the horse population crashed. Suddenly the price of feeding and maintaining a horse was much higher than the alternative. There is no law of economics that says the supply and demand of a good can't fall below the cost of maintaining it. Or that wages can't fall below minimum, in our case.
Robotics has made incredible advancements over the past 50 years or more. They have taken over entire factories, doing countless routine tasks previously done by humans. But they are still very limited. They have 0 intelligence - they can't see, they can't learn. They can only perform a rote series of movements. So there is still tons of work available for humans.
But with recent advancements in machine learning, this is about to change. A robot will soon be able to be trained to flip burgers, or drive a car, or take a customer's order, etc, etc. I can not imagine any jobs that an average, unskilled human can do, that a machine won't soon be able to do. Maybe skilled professions will be protected - I can't imagine robots being able to program computers for awhile. But the vast majority of humans can not be trained to be computer programmers.