I don't understand the reference to drinks. Can you make it clearer? In English we could very well say "He has his lunch." because as you say "have" is commonly used for "eat". There is no hint of a drink or not.
We accept "has" or "is having" lunch for this sentences but as we are trying to teach the verb " τρώω" we show that as the main answer.
Jaye, Thanks for clarifying the choice of words. BTW, the English brunch is more than shifting the time of either meal. It is actually a combination of the two. The https://en.wikipedia.org/wiki/Brunch has a pretty good description of the history and meaning of the term. I enjoy the Greek course very much indeed AND I appreciate your great follow-ups as well! Enjoy the lingot, Daniel.