https://www.fellponymuseum.org.uk/fells/rom_dark/size.htm suggests that horses really WERE shorter. But so were people. And even so, riders were drawn a lot bigger relative to their horses than they are today. Which bones back up.
It is not clear to me how the two trade off back in the days of chariots.
Horses definitely got much bigger but they've been big enough to ride since they were domesticated. The small Mongolian pony is still 5-600 pounds. Plenty big enough to carry ~200 pounds of people + arms.
https://www.fellponymuseum.org.uk/fells/rom_dark/size.htm suggests that horses really WERE shorter. But so were people. And even so, riders were drawn a lot bigger relative to their horses than they are today. Which bones back up.
It is not clear to me how the two trade off back in the days of chariots.