The idea that people eat the relevant animal on its New Year celebrations is a myth. For a start, of the 12 animals of the Chinese year cycle, one is mythical, two are illegal to eat and three unacceptable to most people.
Mythical: Dragon
Illegal to eat: Tiger, Monkey
Unacceptable; Rat, Snake, Dog
This weekend will be my 29th CNY in China and I have been served the same food every one of those years.
The only one of the 12 to maybe appear regularly has been chicken but never to my knowledge rooster.
I have eaten everything on the list except the dragon, tiger and monkey but not at CNY.