You are pulling it away but the heat is still going into the room.Well ! I would think, if I were starting a new fire, and I wanted the fire box to get hot , it would take longer to get it hot, if I am pulling heat away from it with a blower.
But maybe I'm wrong
But I would agree on start up with fans on it would take longer to get the stove to temp for kicking in the cat or the tubes.
But I thought you were saying the wood burns up faster if the fans are on and I don't really think so.
You would have to add more air to the fire .