What I am trying to get after here is how did we get to the point of getting to a certain age, then taking a written exam, then taking an actual driving test before being allowed to drive, as well as if you break too many of the rules, you will be banned from partaking in this form of transportation by the state. I'm sorry I didn't make that clear in OP.
Are there examples of that process before the invention of cars. Did you have to pass a test to ride a horse, pull a wagon, move cattle, drive a train, etc...? If you crashed your horse while riding drunk, could the state prohibit you from riding your horse?
Your argument is invalid.
The idea that there was free movement of walking, horses, and carriages is mistaken entirely.
For example, let's look at this Mississippi Code:
§ 97-25-7. Railroads; driving vehicle or livestock on track
Any person who shall ride, drive any vehicle, drive any cattle, horses, mules or other livestock along or on any railroad track open and operated for traffic, unless by permission of the owners of said track, or their agent, shall be guilty of a misdemeanor and be fined not less than twenty-five dollars ($ 25.00) nor more than two hundred fifty dollars ($ 250.00). The penalty hereof shall not be incurred by operating a street railroad or by crossing a track.
The historical citations and versions go back to 1892
Here's another Mississippi law concerning bridges.
§ 65-7-41. Rate of travel over bridges
It shall not be lawful for any person to drive a wagon, carriage, or other vehicle of any description drawn by animal power or to ride, leap, or drive any horse, mule, or any description of cattle over or upon any bridge erected in any county by authority of the board of supervisors or in a city, town, or village at a faster gait than a walk; and every person so offending shall be guilty of a misdemeanor and, on conviction, fined Five Dollars ($ 5.00). But the board of supervisors or municipal authorities shall cause to be fixed up and kept at each end of the bridge, in some conspicuous place, a board on which shall be painted in large letters, "Five Dollars ($ 5.00) fine for riding or driving any animal over this bridge at a faster gait than a walk"; and on neglect of such notice, such fine shall not be incurred.
This one goes back to 1857.
The idea that there were no regulations of the usage of horses, carriages, barges, ferries, or walking before the invention of cars is absolutely not true. Laws governing the locations of operation, taxation on services, etc., have existed for centuries.
Here is a copy of the General Statues of the Commonwealth of Massachusetts
Pay close attention to page 423. It concerns rights of way, safety features (bells on sleighs), and obstructing roadways.
I unfortunately don't have access to my sources at present, so this is going off memory. Eternalkerri is right that there were some state, local and city ordinances that regulated traffic in the days before the Horseless Carriage, but the modern system of comprehensive "rules of the road" are very much a product of the automobile age. The realities of the horseless carriage - which could travel at much higher speeds than any beast of burden, and unlike a horse (a living creature, capable of acting independently and taking measures to avoid collision regardless of human imput) a car depends entirely on its driver. The modern system of traffic laws can be traced to William P. Eno, whom you can read more about in my post here
Sorry for the brevity, I hope that helps!