I agree the terrain restricted very much the movement of the chariots. Initially Partians as well as the most of the Steppe tribes later initialized hit and run tactics which were very well executed on top of a horse due to their mobility though the archery recieved great bust from the usage of chariots. Chariots were also obsolete for long term invasion, which required precise maneuver through different types of terrain leaving the choice of the battlefield primarily to the invaded country. Indeed, Alexander's march to the East meant more or less the end of the chariots, riding a horse no matter how more difficult it were provided a better deal of surprize battle tactics and for the next two millenia it turned to be the major military force because it could easily take on poorly defended stations and it's high mobility and ability to maneuvre. And while the chariots required usually more than just one rider, the cavalry-men could easily become a more sophisticated force on their own.
About huns' saddles I don't know much, though sometime ago I watched a documentary that suggested they were smaller and elevated the body quite higher than a normal saddle would. They were very well suited to the tactics of the Huns which required a greater form of mobility.