No, they weren't. The disappearance of chariots was largely due to bigger horses which could carry a man being bred. Which made the clunky chariot completely obselete (even then they tended to be used as "battle taxis" rather than as cavalry).Anyway, always remember that roman horses were just like a pony, I mean they were more little than modern horses
The Romans bred good quality mounts in the Rosea Rura, and the Celts, Iberians, Parthians and many others also had good horses.
This is a summary of an article called " The appeal of the ancient horse: a condition of the question applied to the Age of Iron of the Iberian Peninsula" by Quesada, the most important military ancient historian in Spain.
The complete article is in a book which I read a long time ago in Sabadells library. It basically affirms that horses measured 1,30 or 1,40, but they were as good for the battle as bigger horses.
However, in Orient the horses were bigger.
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