In recent years, further improvements in the efficiency of Cu$_2$ZnSn(S,Se)$_4$ photovoltaic devices have been hampered due to several materials issues, including cation disorder. Cu$_2$SnS$_3$ is a promising new absorber material that has attracted significant interest in recent years. However, similar to CZTS, Cu$_2$SnS$_3$ displays cation disorder. In this work, we develop synthetic techniques to control the disorder in Cu$_2$SnS$_3$ thin films. By manipulating the disorder in this material, we observe crystal structure changes and detect improvements in the majority carrier (hole) transport. However, when the minority carrier (electron) transport was investigated using optical pump terahertz probe spectroscopy, minimal differences were observed between the ordered and disordered Cu$_2$SnS$_3$. By combining these results with first-principles and Monte Carlo theoretical calculations, we are able to conclude that even ostensibly "ordered" Cu$_2$SnS$_3$ displays minority carrier transport properties corresponding to the disordered structure. The presence of extended planar defects in all samples, observed in TEM imaging, suggests that disorder is present even when it is not detectable using traditional structural characterization methods. The results of this study highlight some of the challenges to the further improvement of Cu$_2$SnS$_3$-based photovoltaics, and have implications for other disordered multinary semiconductors such as CZTS.