Smaller and piece of something are not the same thing, but I get what you mean. However, this case is completely upside down: rood is a general term for anything edible whereas meal is either a portion of food or a customary time it is being consumed.
OK, it's an exception, I'll have to memorize it but I'm curious about how it came to be in Esperanto which is prised for its regularity and logical word building.
I always thing of aĵ as "stuff".
So manĝaĵo is "meal stuff" or "the stuff that makes up a meal"
I don't think Esperanto really has the reverse of that. Something that takes a generic concept and makes it specific. A meal is a specific unit of food, but food itself is just meal stuff.
On a related note, it really bothers me that the word for "open" is malferma. It just sounds like the positive sounding statement should be negated to produce the negative statement. That makes the whole mal- rule seem arbitrary to me.
The mal- prefix sometimes leads me on in a similar way as well :-). In some languages it's a prefix used for negation whereas Esperanto uses it for inverse. I was surprised at the beginning as well, because I'd assume the root word would be for open and then one could create not-open but well it's closed and not-closed.