"An egg is a food."
Agreed. As a US English speaker, "X is a food" sounds just fine to me. Or at least I'm sure I've heard it before...
But maybe it's incomplete. How about, "an egg is a food that has many uses in cooking." Does that sound more natural?
It would actually sound weird to me to say "a type of food" in that sentence, because it would sound like eggs are a whole category of food, like fruits or vegetables, but really when I say "an egg", I'm referring to only one kind of egg (chicken), and probably so are you.