You’re misreading things. Reread the passage: “a*** behind-the-machine human*** quickly slipped another sandwich, salad, piece of pie or coffee cake into the vacated chamber.”
You could see people working behind the machines (each of the doors had a glass door on the kitchen side, so the staff could see what needed replenishing).
The basic operation was that the behind-the-machine staff would put an item in the compartment (they were designed to keep foot hot or cold as necessary, so you could have hot food like meat loaf) and shut the door. A customer would purchase the item. The staff would see the spot was empty and replace what was there.
No one ever brought food to you. The only people working the floor were the people giving change or those busing tables.
Automats didn’t make their money selling burgers. They had them, but it meant the burgers would sit (unwrapped) in the warming chamber; if there wasn’t a lot of turnover, they were very unimpressive. They made their (food) reputation on sandwiches and desserts and coffee. Their hot entrees would be baked things like meat loaf, and their potatoes were far more likely to be mashed than fried. McDonald’s burgers were wrapped, and the fries better.
In addition, the automats were a strictly big-city phenomenon. McDonald’s was suburban at first (with plenty of parking) and did a better job with fried foods. It was a different niche, but one that burgeoned in the 50s and 60s.
Automats did not have wait staff, except for busing tables. Certainly the only people I remember at the one on 42nd street didn’t have waiters. This was in the '60s.
However there were other Horn and Hardarts with table service, one was the go to place after the movies near my house. I think they had Automat style coffee dispensers but I didn’t drink coffee back then so I never used them.