Looking at art, they drove from the vehicle rarely. Many horses are lead, ground driven, or postillion ridden. One reason for this could have been a lack of comfort due to a lack of modern elliptical leaf springs.
I first encountered the famed coach of Hungary when a blogger praised its unique suspension and its speed. The post referenced the image (right/below) redrawn from a stamp based off a period drawing. The comfortable vehicle carried passengers and mail across great distances. The blog credited unique technologies.
Image by Else
In 1992, a museum in the town of Kocs, Hungary built a reproduction from period accounts. The supporting documentation included both the 1568 Schemel image and a 1518 written account by a passenger.
Freiherr Sigmund von Herberstein (1518) - Google Translate (Lejos, 2007)
We got a car in a village 10 miles from Pest, which was drawn by three horses running side by side, and at the same time iron it wasn't at all. Four people came with the car...
I got on a Hungarian carriage, which was pulled by three horses, ... horses we exchanged ... 5 miles from Győr in the village of Kocs, from which the transport carts they got their name ...
It is difficult and uncomfortable to drive poorly sprung vehicles while seated. Lack of suspension favors standing, leading, and postillion as driving positions. Novel suspension systems used on good roads broaden vehicle utility.