I always thought it was because they were waiting for you to order, waiting for your food to be ready to bring you and then waiting for you to pay (you don't have to leave right away) - hence they are literally "waiting on you", as the phrase goes. As a patron, you are more or less "demanding" that those things all occur at your whim.
Now they call themselves "servers". I like how they come up to your table and say, "Hi, I'm Brittany. I'll be taking care of you today." I want to say, "Ok, Brittany, how about you pay my electric bill and get the tires on my car rotated?"
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