I guess you should look before you leap then >.>
My theory is that Han can "control" steam w/o having chakra affinities in Katon and/or Suiton. Just like how Gaara control sand w/o the hint of him having chakra affinities in Fuuton and/or Doton.
Taking into consideration of the KGs that involve fusion of 2 elements: Youton, Mokuton, Hyouton, Futton, Ranton, Jinton...
The users didnt need to carry a certain amount of their advanced element with them, like you dont see Yamato carrying a large amount of wood with him to perform his Mokuton. There's the exception of Hyouton where the user would need a water source to convert it into ice but then again there are some water ninjutsus where the user can spit water out of his own mouth. However the fact that they show Yamato release a piece of wood from any part of his body is kinda creepy >.>
Going back to Han and Gaara. Why cant Han simply release steam out of his mouth like what 5th Mizukage did? Why cant Gaara spit sand out of his mouth since we seen in the manga that 3rd Hokage can spit out mud to perform Doton: Doryuuheki. This could only mean that Han's steam jutsus and Gaara's sand abilities are NOT a fusion of 2 elements and that they both stand in a league of their own, simply classified as ninjutsu. Gaara needs his gourd to carry sand and Han needs his furance to produce steam.
EDIT: I want to emphasize that this is my theory and not a fact.