You asked us a question. "Why did ancient Greeks and Romans eat lying down?"
Reclining and dining in ancient Greece started at least as early as the 7th century BCE. It was later picked up by the Romans.
They ate lying down while others served them. It was a sign of power and luxury. It was enjoyed by the elite. People further down the social ladder copied the laid-back dining style. If they could afford to.
Who would not want to stretch out while chowing down?
But not everyone was so lucky in ancient Greece. Women did not usually get invited to banquets. Except for rare events like wedding feasts. And even then they had to sit upright.
It was only in ancient Rome that customs changed. They allowed upper-class women to lounge like the men.
It sounds sweet. But all that lying down and eating could not have been good for the heartburn.