One of the main reasons why the exhaust is at the bottom is so that the barrel can then act as a radiator for the heat before continuing into the mass or up and out a chimney.
So no, you really don't have to have it located at the bottom if that fits your goals better. It certainly will add a lot more heat up and out the chimney and therefore increase draft as a result but if your fine with that then by all means go for it.
One final thought: Some people do both. They have the main exhaust at the bottom and a bypass near the top which can be opened and closed at will to either help start a cold stove or just don't want the radiant heat in the room as would be the case for an indoor cooking stove in summer.