Indeed, the current scholarly consensus, according to all the best books about food in ancient Greece, is that flesh meat was seldom eaten except as part of a sacrifice - which were by no means all major state occasions. A sacrifice could be arranged by anyone who could pay for the animal and for the mageiros, the specialist sacrificer and cook who did the hard work.
I'm adding here a description of a sacrifice from my short historical novel, The Priestess and the Slave. The 5th century BC narrator has suffered a stillbirth that nearly killed her; a year later, her husband is celebrating the birth of a healthy baby to their slave Glukera.
... A few days later, he held the proper feast for all his friends and relatives, even my parents, hiring a man to sacrifice a full-grown sheep. After the sacrificer-cook slit the sheep's throat and let the beast's blood pour over the small altar in our courtyard, he sliced up the animal on the table he'd brought with him, removed the skin, and wrapped long bones and other morsels in fat to burn on the altar as the gods' rightful portion. We prayed that all the gods would enjoy the smoke, as the sacrificer-cook poured unmixed wine over the flames.
Next, the man cut up the best of the animal's insides, the heart, the spleen, the kidneys and the tender liver, and slid them onto iron skewers. They went on the altar to roast as special treats for the guests of honor. While they were on the flames, the sacrificer-cook skillfully butchered the remaining carcass, cutting the muscle meat into chunks and setting them to boil in a tripod cauldron, with onions, herbs and barley. Slaves mixed great bowls of wine with water, and served it out to all, while we waited for the meat to stew.
I was not sad, during this merriment. The gods had willed my childless state; it seemed to fit, somehow, with my odd ways of thought. I'd been a distant, unsatisfactory daughter to my mother, a constant source of discontent to her. Perhaps it was better that one like me did not bring up a child. Perhaps . . .
Glukera sat her baby boy on my childless lap, and his starfish hands grasped for my empty breasts. I gently disengaged his tiny hands and gave him my gold bangle instead. He made a soft noise, bringing the bright bangle to his mouth, and Glukera smiled at me.
I nodded; it was good. When Kinesias grew to adulthood, he and his future wife would tend the family graves, giving the dead the oil, milk and honey that was their due, as I did now. My own dead baby would not go hungry, down there in the dark earth.